Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language: Applying Algebraic Modelling Language Techniques to Integrated ... Systems (Green Energy and Technology) 3030003949, 9783030003944

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Green Energy and Technology

Nnamdi Nwulu Saheed Lekan Gbadamosi

Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language

Green Energy and Technology

Climate change, environmental impact and the limited natural resources urge scientific research and novel technical solutions. The monograph series Green Energy and Technology serves as a publishing platform for scientific and technological approaches to “green”—i.e. environmentally friendly and sustainable—technologies. While a focus lies on energy and power supply, it also covers “green” solutions in industrial engineering and engineering design. Green Energy and Technology addresses researchers, advanced students, technical consultants as well as decision makers in industries and politics. Hence, the level of presentation spans from instructional to highly technical. **Indexed in Scopus**.

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Nnamdi Nwulu Saheed Lekan Gbadamosi •

Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language

123

Nnamdi Nwulu Department of Electrical and Electronic Engineering Science University of Johannesburg Johannesburg, South Africa

Saheed Lekan Gbadamosi Department of Electrical and Electronic Engineering Science University of Johannesburg Johannesburg, South Africa

ISSN 1865-3529 ISSN 1865-3537 (electronic) Green Energy and Technology ISBN 978-3-030-00394-4 ISBN 978-3-030-00395-1 (eBook) https://doi.org/10.1007/978-3-030-00395-1 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Today’s modern grids are complex interconnected systems with increased customer participation and intricate ancillary services. The incorporation of modern technologies including renewable energy sources and electric vehicles have increased the complexity of modern power systems. Furthermore, today’s power system requires high reliability as brownouts and blackouts have tremendous economic, social, environmental, and political implications. This has led to increased focus on how to optimally operate, monitor and control power systems. Thus, this book Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language focuses on the use of Algebraic Modelling Languages as a simplified approach for modelling and solving complex mathematical optimisation problems found in modern power grids. The Advanced Interactive Multidimensional Modelling System (AIMMS) is the AML used in this book. This comprehensive text for power system operations is divided into three parts. The first part is an introductory part and spans Chaps. 1–3. In Chap. 1, an introduction and overview of the electrical power system is given. It covers the major components of a power system and reviews the various renewable energy sources (RES) that constitute today’s energy mix. The chapter also delves into smart grids and considers various definitions of a smart grid. The deregulation of the electric power industry across nations of the world is also considered and the chapter concludes with operational and control issues in a power system. Chapter 2 reviews the concept of demand side management (DSM) programs, which aim to foster a more responsible use of electrical energy. DSM encompasses both energy efficiency and demand response (DR) programs. Examples of energy efficiency programs (a retrofit program), price-based DR (time of use appliance scheduling) and IB-DR programs (game theory-based incentive-based DR) were discussed in detail and their mathematical formulations given. In Chap. 3, mathematical optimization modelling is considered. The two broad classifications of mathematical modelling solution approaches, classical and meta-heuristic methods, are detailed and various examples are given. The chapter also gives practical steps on how to model mathematical problems. Furthermore, v

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algebraic modelling languages (AMLs) are considered and AIMMS, which is a prominent AML, is introduced. Finally, examples were shows of different energy efficiency and DR programs and their notation in AIMMS. The second part details various power system operations and control models including demand response programs and spans Chaps. 4–10. The AIMMS source code for all the models is included. Chapter 4 considers the Dynamic Economic Emissions Dispatch (DEED) problem which is concerned with assigning customer loads to the available power-generating units in order to ensure a reliable and secured system whilst minimizing fuel cost and emissions. In this chapter, different variations of the DEED problems were investigated. The variations considered include the Dynamic Economic Dispatch (DED), Emissions Constrained Dynamic Economic Dispatch (ECDED), Bid Based Dynamic Economic Emissions Dispatch (BBDEED) and the Profit Based Dynamic Economic Emissions Dispatch (PBDEED). An Incentive Based DR program is also incorporated into the DEED problem-giving rise to the IBDEED problem and is considered in this chapter. All models were solved using AIMMS with source codes given. In Chap. 5, the Generator Maintenance Scheduling problem was presented. Specifically, the Reliability Constrained GMS (RC-GMS), which seeks to minimize the sum squares of reserve of the generating units without violating maintenance window constraints is developed and solved using AIMMS. Chapter 6 considers the Combined Heat and Power Dispatch (CHPD), which obtains the optimal dispatch of combined heat and power generators whilst minimizing daily operational cost and satisfying users power and heat demand throughout the total dispatch interval. Variants like the Combined Heat and Power Dynamic Economic Dispatch (CHPDED), Combined Heat and Power Pure Dynamic Emissions Dispatch (CHPPDED), Combined Heat and Power Dynamic Economic Emissions Dispatch (CHPDEED) and Profit-based Combined Heat and Power Dynamic Economic Emissions Dispatch (PBCHPDEED) are considered. Again, an incentive-based DR program is integrated into the CHPDEED problem giving rise to the IB-CHPDEED formulation. All models are solved using AIMMS and the source code given. In Chap. 7, a grid-connected hybrid renewable energy powered microgrid incorporating a DR program is presented. The main aim is to minimize the operational cost of the fossil fuel generators and the cost of transferrable power traded between the main grid and microgrid, as well as maximizing the incentive based DR. Another variant of the model but including battery storage systems is also detailed. Both models are modelled and solved with AIMMS with relevant source codes detailed. Chapter 8 considers the Optimal Power Flow whose objective is to minimize the operational cost of the power system while maintaining effective generation scheduling with efficient power flow on the transmission lines. The Direct Current OPF is the variant of OPF modelled in this chapter and the AIMMS code and results detailed.

Preface

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In Chap. 9, the Transmission Network Expansion Planning (TNEP) is presented. TNEP aims at determining the optimal approach for adequate selection of transmission lines to be constructed to meet the growing energy demand, as well as facilitating power transfer from far-away locations to consumers. Two variants of TNEP: The static and dynamic TNEP are modelled and the incorporation of demand response programs into the TNEP problem is presented with the relevant AIMMS code detailed. Chapter 10 gives detail of the GCEP problem formulation. The chapter considers different planning approaches, ranging from load forecasting, real and reactive power capacity to DR integrated GCEP. The GCEP models were formulated for both DC and AC planning methods, as well as real and reactive power capacity. Different examples are illustrated in this chapter and their solutions are provided with appropriate AIMMS codes. The final part consists only of Chap. 11, which details the conclusions of the book whilst highlighting possible future extensions and considerations. Johannesburg, South Africa

Nnamdi Nwulu Saheed Lekan Gbadamosi

Contents

1

Power Systems and Renewable Energy Systems 1.1 Introduction and Overview . . . . . . . . . . . . 1.1.1 Generation of Electric Power . . . . 1.1.2 Transmission of Electric Power . . . 1.1.3 Distribution of Electric Power . . . . 1.1.4 Hydropower . . . . . . . . . . . . . . . . . 1.1.5 Solar Energy . . . . . . . . . . . . . . . . 1.1.6 Wind Energy . . . . . . . . . . . . . . . . 1.1.7 Bioenergy . . . . . . . . . . . . . . . . . . 1.1.8 Geothermal Energy . . . . . . . . . . . . 1.1.9 Ocean Energy . . . . . . . . . . . . . . . 1.2 Smart Grids . . . . . . . . . . . . . . . . . . . . . . . 1.3 Electricity Deregulation . . . . . . . . . . . . . . . 1.4 Operation and Control . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Demand Side Management . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Energy Efficiency Programs . . . . . . . . . . . . . . . . . . . . 2.2.1 Building Retrofit Energy Efficiency Programs 2.3 Demand Response Programs . . . . . . . . . . . . . . . . . . . 2.3.1 Price-Based Demand Response . . . . . . . . . . . 2.4 Incentive-Based Demand Response . . . . . . . . . . . . . . 2.4.1 Direct Load Control . . . . . . . . . . . . . . . . . . . 2.4.2 Interruptible Programs . . . . . . . . . . . . . . . . . 2.4.3 Curtailable Load Program . . . . . . . . . . . . . . . 2.4.4 Demand-Bidding Programs . . . . . . . . . . . . . . 2.4.5 Game Theory Demand Response Programs . .

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2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Mathematical Optimization Modeling and Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classical Mathematical Optimization Solution Approaches . 3.2.1 Gradient Descent-Based Methods . . . . . . . . . . . . . 3.2.2 Direct Search Methods . . . . . . . . . . . . . . . . . . . . . 3.3 Meta-Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Particle Swarm Optimization . . . . . . . . . . . . . . . . . 3.3.3 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Group Search Optimization . . . . . . . . . . . . . . . . . . 3.3.5 Cuckoo Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Grey Wolf Optimization . . . . . . . . . . . . . . . . . . . . 3.3.7 Harmony Search Algorithm . . . . . . . . . . . . . . . . . 3.3.8 Bee Colony Optimization . . . . . . . . . . . . . . . . . . . 3.4 The Mathematical Modeling Process . . . . . . . . . . . . . . . . . 3.5 Algebraic Modeling Languages . . . . . . . . . . . . . . . . . . . . . 3.5.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Advanced Interactive Multidimensional Modeling System . . 3.6.1 Software Installation . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 AIMMS Tutorials . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Building Retrofit Energy Efficiency Programs Using AIMMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Appliance Scheduling Using Time-of-Use Demand Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Incentive-Based Demand Response Modeling Using AIMMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Economic Emissions Dispatch . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Dynamic Economic Dispatch . . . . . . . . . . . . . . 4.1.2 Dynamic Economic Emissions Dispatch . . . . . . 4.2 Emission-Constrained Dynamic Economic Dispatch . . . . 4.3 Bid-Based Dynamic Economic Emission Dispatch . . . . . 4.4 Incentive-Based Dynamic Economic Emissions Dispatch 4.5 Profit-Based Dynamic Economic Emissions Dispatch . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generator Maintenance Scheduling . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.2 Reliability Criterion Generator Maintenance 5.2.1 Formulation of RC-GMS Model . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Combined Heat and Power Dispatch . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Combined Heat and Power Dynamic Economic Dispatch . . . 6.3 Combined Heat and Power Pure Dynamic Emissions Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Combined Heat and Power Dynamic Economic Emissions Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Incentive-Based Combined Heat and Power Dynamic Economic Emissions Dispatch . . . . . . . . . . . . . . . . . . . . . . . 6.6 Profit-Based Combined Heat and Power Dynamic Economic Emissions Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hybrid Grid-Connected Renewable Energy Sources Powered Microgrid with Demand Response . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Hybrid Renewable Energy Sources . . . . . . . . . . . . . . . . . 7.2.1 Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Solar Photovoltaic . . . . . . . . . . . . . . . . . . . . . . . 7.3 Microgrid Storage Systems . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Battery Storage System . . . . . . . . . . . . . . . . . . . . 7.3.2 Pumped Storage Hydro System . . . . . . . . . . . . . . 7.3.3 Flywheel Storage System . . . . . . . . . . . . . . . . . . 7.3.4 Supercapacitor Storage System . . . . . . . . . . . . . . 7.3.5 Hybrid Storage System . . . . . . . . . . . . . . . . . . . . 7.4 Hybrid Renewable Energy Sources with Battery Storage in a Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Incentive-Based Hybrid Renewable Energy Sources Powered Microgrid with Demand Response . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimal Power Flow . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . 8.2 DC Optimal Power Flow . . . . . . . . . . . . 8.2.1 Formulation of DC-OPF Model . 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . 8.4 Nomenclature . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Transmission Network Expansion Planning . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Static Transmission Network Expansion Planning . . . 9.2.1 DC Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 AC Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . 9.3 Dynamic Transmission Network Expansion Planning (DTNEP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 DC Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . 9.4 Demand Response Integrated Dynamic Transmission Network Expansion Planning . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Generation Capacity Expansion Planning . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Load Forecasting . . . . . . . . . . . . . . . . . . . . . . . 10.3 Real Power Capacity Expansion Planning . . . . . 10.3.1 Static Model . . . . . . . . . . . . . . . . . . . . 10.3.2 Nomenclature . . . . . . . . . . . . . . . . . . . . 10.4 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Nomenclature . . . . . . . . . . . . . . . . . . . . 10.5 Reactive Power Capacity Expansion Planning . . 10.5.1 Nomenclature . . . . . . . . . . . . . . . . . . . . 10.6 Demand Response Integrated Generator Capacity Expansion Planning . . . . . . . . . . . . . . . . . . . . . 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 11.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

List of Figures

Fig. Fig. Fig. Fig.

1.1 1.2 1.3 1.4

Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 2.1 Fig. 2.2 Fig. 3.1 Fig. 3.2 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

3.3 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Basic structure of a power system . . . . . . . . . . . . . . . . . . . . . World electricity generation from RES and non-RES . . . . . . . Total world electricity consumption . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation using hydropower (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation using solar energy (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation using wind (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation from bioenergy (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation using geothermal energy (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldwide electricity generation using ocean/marine energy (2000–2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration of RES integrated into a microgrid . . . . . . . . . . Overview of a smart grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . An overview of DSM, energy efficiency and demand response programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classification of DR programs . . . . . . . . . . . . . . . . . . . . . An overview of mathematical optimization techniques . . . . . . An overview of meta-heuristic optimization solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow chart for group search optimization . . . . . . . . . . . . . . . . Total hourly load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal capacity of generating units . . . . . . . . . . . . . . . . . . . . Total hourly load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total hourly load for emission-constrained DED model . . . . . IEEE 30-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Output power generated by the generators . . . . . . . . . . . . . . . Power curtailed by the customers . . . . . . . . . . . . . . . . . . . . . .

.. .. ..

2 2 3

..

5

..

7

..

8

..

8

..

9

.. .. ..

10 11 13

.. .. ..

22 25 39

. . . . . . . . .

41 43 65 68 69 76 78 88 89

. . . . . . . . .

xiii

xiv

List of Figures

Fig. 4.8 Fig. 4.9 Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.10 5.1 5.2 5.3 7.1 7.2 7.3

Fig. Fig. Fig. Fig. Fig.

7.4 7.5 7.6 9.1 9.2

Fig. 10.1

Incentive paid to the customers . . . . . . . . . . . . . . . . . . . . . . . . The forecast energy price for the profit-based DEED model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total hourly load for the profit-based DEED model . . . . . . . . Crew availability for maintenance . . . . . . . . . . . . . . . . . . . . . . Total available generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reserve margin and maintenance capacity for GMS . . . . . . . . PV power generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind power generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power produced by PV and battery (charging and discharging) system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind and solar power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power output from fossil fuel generators . . . . . . . . . . . . . . . . Incentive and power curtailed by customers . . . . . . . . . . . . . . IEEE Garver six-bus system network . . . . . . . . . . . . . . . . . . . Optimal power flowing on the existing and prospective transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified IEEE Garver six-bus system network . . . . . . . . . . .

..

89

. . . . . . .

. . . . . . .

92 92 105 105 105 161 161

. . . . .

. . . . .

161 169 170 170 188

. . 191 . . 216

List of Tables

Table 1.1 Table Table Table Table Table Table

4.1 4.2 4.3 4.4 4.5 4.6

Table Table Table Table

4.7 4.8 4.9 4.10

Table Table Table Table Table Table Table

4.11 4.12 4.13 4.14 4.15 4.16 4.17

Table Table Table Table Table Table Table Table Table

5.1 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Overview of electricity deregulation in organization for economic cooperation and development countries . . . . . . . Generator data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal generated power for DED model . . . . . . . . . . . . . Generator data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal generated power for DEED model . . . . . . . . . . . . Generator data for emission-constrained DED model . . . . . Optimal generated power for emission-constrained DED model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generator data for bid-based DEED model . . . . . . . . . . . . Data for customer bid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for optimal bids . . . . . . . . . . . . . . . . . . . . . . . . . . . Generator and customer bid costs for IEEE 30-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters for the six generators . . . . . . . . . . . . . . . . . . . . Hourly load demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power produced by generators . . . . . . . . . . . . . . . . . . . . . . Power curtailed by customer . . . . . . . . . . . . . . . . . . . . . . . Customer incentive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generator data for profit-based DEED model . . . . . . . . . . . Optimal generated power for the profit-based DED model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data for the 21-generator units test system . . . . . . . . . . . . . Data for thermal-generating units . . . . . . . . . . . . . . . . . . . . Data for CHP-generating units . . . . . . . . . . . . . . . . . . . . . . Data for heat-generating unit . . . . . . . . . . . . . . . . . . . . . . . Data for heat and power demand . . . . . . . . . . . . . . . . . . . . Optimal power flow for CHPDED model . . . . . . . . . . . . . . Optimal heat flow and losses for CHPDED model . . . . . . . Optimal power flow for CHPPDED model . . . . . . . . . . . . . Optimal heat flow for CHPPDED model . . . . . . . . . . . . . .

. . . . . .

. . . . . .

16 64 65 69 72 73

. . . .

. . . .

76 78 81 81

. . . . . . .

. . . . . . .

82 84 84 85 86 87 91

. . . . . . . . . .

. . . . . . . . . .

93 99 112 112 112 112 113 118 123 124 xv

xvi

List of Tables

Table Table Table Table Table Table Table Table Table Table

6.9 6.10 6.11 6.12 6.13 6.14 7.1 7.2 7.3 7.4

Table Table Table Table

7.5 7.6 7.7 7.8

Table Table Table Table Table Table

8.1 8.2 8.3 8.4 9.1 9.2

Table 9.3 Table 9.4 Table 9.5 Table 9.6 Table 9.7 Table 9.8 Table 9.9 Table 9.10 Table 9.11 Table 9.12 Table 9.13 Table 9.14 Table 9.15 Table 10.1

Optimal power flow for CHPDEED model . . . . . . . . . . . . Optimal heat flow and losses for CHPDEED model. . . . . . Optimal power flow for IBCHPDEED model . . . . . . . . . . . Optimal heat flow and losses for IBCHPDEED model . . . . Optimal power flow for PBCHPDEED model . . . . . . . . . . Optimal heat flow and losses for PBCHPDEED model . . . Data for conventional generator . . . . . . . . . . . . . . . . . . . . . Technical specifications for the RES . . . . . . . . . . . . . . . . . Data for the load demand . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power flow for the conventional and hybrid RES generating units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data for conventional generator . . . . . . . . . . . . . . . . . . . . . Hourly power demand and power interruptible values . . . . Hourly power forecast from wind and solar generation . . . Power generated from fossil fuel generators and transfer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters for IEEE 14-bus test system . . . . . . . . . . . . . . . Optimal power generated from the generators . . . . . . . . . . Bus voltage angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power flow between two buses . . . . . . . . . . . . . . . Data for IEEE Garver six-bus system example . . . . . . . . . . Parameters and capacities of the existing and prospective transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal generated power from the generators for DC STNEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power flow on the existing and prospective transmission lines for DC STNEP model . . . . . . . . . . . . . . Voltage phase angle at the buses for DC STNEP model . . Real and reactive parameters for IEEE Garver six-bus system example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real and reactive power for AC STNEP model . . . . . . . . . Optimal power flow and line losses on the existing and prospective transmission lines for AC STNEP model . . . . . Voltage magnitude and phase angle for AC STNEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified load data for 10-year planning horizon . . . . . . . . Optimal generated power for DC DTNEP model . . . . . . . . Optimal power flow on the transmission lines for DC DTNEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal generated power for DTNEP-DR model . . . . . . . . Optimal power flow on the lines for DTNEP-DR model . . Optimal load curtailment for 100% participating factor for DTNEP-DR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified data for IEEE Garver six-bus system example . .

. . . . . . . . .

. . . . . . . . .

130 131 139 139 147 148 157 157 157

. . . .

. . . .

160 164 164 165

. . . . . .

. . . . . .

168 180 181 181 182 188

. . 188 . . 191 . . 191 . . 191 . . 194 . . 198 . . 198 . . 199 . . 201 . . 204 . . 204 . . 210 . . 211 . . 211 . . 216

List of Tables

Table 10.2 Table 10.3 Table 10.4 Table 10.5 Table 10.6 Table 10.7 Table 10.8 Table 10.9 Table 10.10 Table 10.11 Table 10.12 Table 10.13 Table 10.14 Table 10.15 Table 10.16 Table 10.17 Table 10.18 Table 10.19 Table 10.20

xvii

Parameters for transmission lines . . . . . . . . . . . . . . . . . . . . Optimal generated power from fossil fuel generators for static DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . Optimal generated power from prospective RES for static DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power flow on the transmission lines for static DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage phase angle at the buses for static DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified load profile for dynamic DC GCEP model . . . . . Optimal power from fossil fuel generators for dynamic DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power from renewable energy generators for dynamic DC GCEP model . . . . . . . . . . . . . . . . . . . . . . Optimal power flow on the transmission lines for dynamic DC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage phase angle for dynamic DC GCEP model . . . . . . Real and reactive parameters for IEEE Garver six-bus system example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal real and reactive power from fossil fuel generators for AC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal real and reactive power from renewable energy generators for AC GCEP model . . . . . . . . . . . . . . . . . . . . . Optimal power flow and line losses on the transmission lines for AC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . Voltage magnitude and phase angle for AC GCEP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power from fossil fuel generators for dynamic GCEP-DR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power from renewable energy generators for dynamic GCEP-DR model . . . . . . . . . . . . . . . . . . . . . . Optimal power flow on transmission lines for dynamic GCEP-DR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal load curtailment for 100% participating factor for dynamic GCEP-DR model . . . . . . . . . . . . . . . . . . . . . .

. . 216 . . 220 . . 220 . . 221 . . 221 . . 227 . . 228 . . 229 . . 229 . . 229 . . 231 . . 240 . . 240 . . 241 . . 241 . . 249 . . 250 . . 250 . . 250

Chapter 1

Power Systems and Renewable Energy Systems

1.1

Introduction and Overview

Electric power systems, sometimes simply referred to as power systems, can be defined as a network of electrical devices that are used to generate, transmit, and distribute electrical power. From this definition, it can be stated that electric power systems have three key roles: the generation, transmission, and distribution of electrical power. Thus, power systems can be classified using these three critical categories. Figure 1.1 gives the basic components of a power system.

1.1.1

Generation of Electric Power

The generation of electrical power or electricity occurs via the conversion of mechanical energy to electrical energy. The devices used for this are often referred to as electromechanical generators. They work by converting kinetic energy and forcing a generator to rotate, thus generating electric power. They can be powered either by thermal processes or by the flow of water or wind. Thermal processes typically use coal and natural gas as the primary fuel. Electricity is typically generated at low voltage and stepped up to be transmitted using the transmission lines. Most of the thermal-powered energy sources described above are referred to as non-RES. They are termed non-renewable because they cannot be replenished in a manner commensurate with consumption. Non-RES are typically fossil fuels. Fossil fuels are those formed from the remains of organisms that lived thousands of years ago. These fuels are a large contributor to harmful atmospheric emissions. RES are energy sources that can be replenished in a manner commensurate with consumption. These energy sources do not release harmful atmospheric emissions and thus are increasingly being used to generate electrical energy. Examples of this kind © Springer Nature Switzerland AG 2021 N. Nwulu and S. L. Gbadamosi, Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language, Green Energy and Technology, https://doi.org/10.1007/978-3-030-00395-1_1

1

2

1 Power Systems and Renewable Energy Systems

Fig. 1.1 Basic structure of a power system [1]

Fig. 1.2 World electricity generation from RES and non-RES [2]

of source include solar, wind, biogas, and geothermal sources etc. Figure 1.2 shows the electricity generation of different energy sources from 1990 to 2016, inclusive of RES and non-RES. Figure 1.3 shows the total world electricity consumption in the same period.

1.1.2

Transmission of Electric Power

The electric power obtained from generating plants is transferred to distribution points in the power system. The network that is used for this long-distance transmission of power is called the transmission network. The transmission network is the interface between the generating stations and the distribution points in the power system. In view of the importance of transmission networks, these are often

1.1 Introduction and Overview

3

Fig. 1.3 Total world electricity consumption [3]

built in such a way that multiple channels exist for power to flow from generating stations to loads in a bid to minimize disruptions. Because of the long distance most transmission networks span, the transmission network has to transmit power at high voltages in order to minimize losses in the system. Transmission networks are usually constructed to support three-phase high-voltage alternating current (HVAC) power. Special kinds of transmission lines are constructed to transmit direct current (DC). These are referred to as high-voltage DC (HVDC) transmission lines. HVDC lines are typically suitable to transfer electric power over very long distances. HVDC lines are efficient and yield lower power losses than other types. HVAC and HVDC transmission lines are usually deployed as overhead lines rather than underground. The choice of overhead construction lines is motivated by the fact that they are cheaper to maintain, even though they are more expensive to construct. Transmission lines’ voltages are in the range of >66 kV with sub-transmission voltage levels in the range of 66–132 kV.

1.1.3

Distribution of Electric Power

The distribution system is the last segment of the electric power supply process. Here electric power is conveyed from the transmission system to the end users. At the distribution level, voltage levels range from 2 to 33 kV. This voltage level is referred to as the distribution voltage level. These voltage levels are further stepped down to between 240 and 415 V, which is the range typically used by households and businesses. This is referred to as utilization voltage. Industrial consumers are often supplied at the distribution voltage level. The incorporation of generators at this spectrum of the power system, placing the generators closer to the end users, is referred to as distributed generation (DG). The incorporation of DG close to the consumer has a number of advantages, chiefly the fact that these generators improve

4

1 Power Systems and Renewable Energy Systems

the reliability and quality of supply. When these DG sources are renewable, they also yield significant environmental benefits. Generators in the power system can therefore be incorporated in two key ways in the power system. The first is at the generation spectrum of the power system. These power sources are large and need to be transmitted across long distances in order to be used by consumers. On the other hand, DG enables the energy sources to be sited close to the consumer, thereby minimizing power losses and improving grid reliability. At both spectrums of the grid, RES can be used. RES are sources of energy that can be readily replenished. Unlike non-RES, they have the added advantage of being environmentally friendly and sustainable. A brief description of prominent RES [4] is detailed below:

1.1.4

Hydropower

Hydropower can be simply defined as power obtained from the kinetic energy of water. This power has historically been used to power mechanical machines such as mills. When energy stemming from fast-flowing water is used to produce electricity, it is referred to as hydroelectricity. Hydroelectricity is considered as renewable energy. Hydroelectricity does not release harmful emissions into the atmosphere even though it has environmental impacts. Hydropower is used in three principal ways in the electricity generation process. The predominant method is the use of a reservoir created by a dam, from which water is released and used to turn turbines and generate power. The second method is the “run of the river” approach [4], where water is steered away from a river into a turbine and electricity is generated directly. The final approach is the pumped hydro storage mechanism. In this approach, two reservoirs are created at different elevations. When the demand for electricity is low, water is pumped from the lower reservoir to the higher reservoir. The water in the higher reservoir is later released to generate power when the demand for electricity is high and more electricity needs to be generated. Figure 1.4 indicates the worldwide use of hydropower.

1.1.5

Solar Energy

Solar energy entails the use of the sun’s radiation to heat either space or water or to produce electricity. As a heating resource, solar energy can be either a passive or active heating source. As a passive heating source for homes, homes are constructed in such a way that sunlight can stream in during the day for space heating and floors are constructed with either dark tiles or bricks that will store heat in the daytime and diffuse it at night [4]. The homes are kept cool in summer by the dark bricks or tiles, which remain cool, and awnings are used to shade the windows from excessive sunlight. Solar energy as a passive heating source, coupled with energy-efficient

1.1 Introduction and Overview

5

Fig. 1.4 Worldwide electricity generation using hydropower (2000–2018) [5]

implementations in a home (energy-efficient appliances and building fixtures) have the potential to minimize energy consumption [4], which will yield a commensurate reduction in emissions. The sun is also a good resource for water heating. Active solar heating works by constructing collectors, which are aligned to face the north and heats water either directly or through a heat transfer fluid. The water is then stored in a water tank and dispersed as in a typical water-heating system. The quantity of hot water produced by a solar water heater is contingent on a number of factors, including the amount of solar radiation, the type of water-heating system, the size of the solar system, orientation and tilt angle of the collectors, among others. There are essentially three types of solar water collectors [4], which define the types of solar water-heating systems. The first type, the most common type of solar water-heating collector, is the flat plate collector, which is made up of an impermeable transparent cover with a dark absorber plate underneath. The second type of collector is the evacuated tube collector, which is a collection of transparent glass tubes arranged in parallel. Each transparent glass tube is made up of the outer glass tube and an internal tube-like absorber. The internal tube is termed an absorber because it absorbs energy from the sun without corresponding heat loss. The third type of collector is the concentrating collector, which works by optically concentrating solar energy prior to its conversion to heat. This type of collector consists of a trough-like concentrator, which transfers direct beam radiation onto the receiver and a receiver component, which converts the absorbed solar radiation into some other form of energy. The flat plate collector, evacuated tube collector and concentrated collector all heat water either as passive solar water-heating systems or active solar water-heating systems [4]. An active solar heating system circulates household water or heat transfer liquids through the solar collector to the water tank via an electric pump or other electric component. A passive solar heating system also circulates water to the tank from the collector, but does not use any electric

6

1 Power Systems and Renewable Energy Systems

component and achieves the water flow via gravity stemming from natural convection. Passive solar water-heating systems are generally more reliable owing to the absence of electric components, while active solar water-heating systems are generally more efficient, albeit expensive. A further distinction between the systems is that in colder climes, the passive water-heating systems tend to freeze, which does not happen to active water-heating systems [4]. Active water-heating systems are also typically hybridized with solar panels in the event of power failures so that water can still be heated. Solar energy can also be used to produce electricity. The use of rays from the sun to generate electricity can be done in two principal ways, either via photovoltaic (PV) technology or solar thermal electric systems. PV solar systems contain semiconducting modules that convert solar radiation to produce DC electricity, inverters to convert the DC to alternating current and often battery storage systems to store excess generated electricity to be used at times when the generated electricity is insufficient to satisfy demand. Auxiliary equipment in PV systems can include the PV hosting structure and protection equipment. The PV panels can be installed on top of buildings, i.e. rooftop power systems, and in general are efficient electricity providers. In essence, PV solar systems convert the sun’s radiation to electricity. Solar thermal electricity-generating systems convert the sun’s radiation to heat, which is then used in turn to generate electricity. The most common solar thermal electricity-generating system is the concentrated solar power (CSP) system. CSP generates electricity by using mirrors to “concentrate” the sun’s rays onto a receiver, which uses the heat to power a conventional generator to generate electricity. There are three types of CSP systems, namely the parabolic trough, power receivers, and dish Stirling systems [4]. The parabolic trough system uses parabolic-like mirrors to concentrate the sun’s energy onto an oil-filled receiver, which runs along the trough’s axis. When the oil is sufficiently hot, it is used to boil water that is deployed in a conventional steam generator for electricity generation. The power receivers consist of a large field of mirrors, which concentrate the sun’s rays onto the top of a tower where the receiver is sited. The generated heat is used to heat molten salt that flows through the receiver, which is in turn used to generate electricity via a conventional steam generator. The dish Stirling system uses a dish-shaped mirror whose surface concentrates the sun’s heat onto a receiver sited at the dish’s focal point. The generated heat from the receiver is transferred to a Stirling engine, which is then able to generate electricity [4]. PV systems can produce electricity on either a small or large scale, while CSP systems (solar thermal systems) typically produce electricity on a large scale. Solar electricity-generating systems, especially when they form part of the RES in a microgrid, can be connected in the off-grid mode, grid-connected mode, or hybrid mode. The off-grid mode is one in which the RES only supplies power and there is no connection from the main grid. When there is grid supply in addition to the RES, it is referred to as grid-connected mode, while the hybrid mode is one with an off-grid solar system coupled with utility power or grid-connected solar power with backup battery storage (Fig. 1.5).

1.1 Introduction and Overview

7

Fig. 1.5 Worldwide electricity generation using solar energy (2000–2018) [5]

1.1.6

Wind Energy

Wind electricity generation uses wind as the prime mover of wind turbines to generate electricity. Wind turbines are electric power generators that convert the wind’s kinetic energy to electrical power. They can rotate along either a vertical or horizontal axis. Horizontal axis wind turbines are more commonly found than vertical axis wind turbines. The wind turbine blades spin around a central hub, which is then connected to a shaft that is the prime mover of a mechanical drive system using an electric generator to generate electricity. A collection of wind turbines is called a wind farm and this can be spread across a large expanse of land. Wind farms can be sited either onshore or offshore. Onshore wind farms are cheaper to construct than offshore wind farms [6]; however, because wind speeds are stronger offshore, offshore wind farms can potentially generate more electricity. Wind turbines, just like PV systems, can be used to generate electricity for small-scale and large-scale uses and can be connected to the grid (Fig. 1.6).

1.1.7

Bioenergy

Bioenergy is energy obtained from biomass. Biomass is organic matter and can originate from plants, animals, or waste products. Bio-power entails generation of power from biomass. This can be done by converting biomass to biogas, which produces methane; the biogas is then used as fuel in a gas turbine and electricity is generated. The conversion of biomass to biogas typically produces methane and carbon dioxide (CO2) as gases. Biomass can also be converted to liquid, which can serve as a fuel. This is often referred to as biofuel [4]. The most popular biofuel is ethanol,

8

1 Power Systems and Renewable Energy Systems

Fig. 1.6 Worldwide electricity generation using wind (2000–2018) [5]

which is obtained by fermenting biomass with a high carbohydrate content. Biofuels have high energy density and have increasingly found industrial application [4]. Biomass can also be converted to pyrolysis oil [4], which can be used for power generation, among other uses. Pyrolysis is obtained from the heating of biomass in a reactor at very high temperatures in the absence of oxygen. The oil that is obtained is then cooled. Another liquid fuel to which biomass can be converted is called biodiesel, which is obtained from animal and vegetable fats. This biofuel has also found industrial application as either a vehicle fuel or fuel additive, as it emits fewer greenhouse gases than conventional vehicle fuels. A further use of biomass in the power-generation process is its integration with coal in coal-fired power stations. Incorporating biomass into the coal feed stock reduces the harmful emission of greenhouse gases, which reduces the environmental impact of burning coal and leads to better air quality [4] (Fig. 1.7).

Fig. 1.7 Worldwide electricity generation from bioenergy (2000–2018) [5]

1.1 Introduction and Overview

1.1.8

9

Geothermal Energy

Geothermal energy is energy hidden in the earth’s core. Temperatures in the earth’s core can be as high as 4982.2 °C and the heat from this core forms pools or reservoirs of water and steam [4]. This heat can then be leveraged for electricity generation and used to satisfy heating and cooling demands. For electricity generation, geothermal power plants are used. These power plants work by obtaining access to the underground steam/water in the earth’s core via a well and using this to power a conventional steam/water turbine, thus generating electricity [4]. The water is usually returned to the earth’s core, replenishing the reservoir and mitigating losses. There are three principal types of geothermal plants, depending on the type of reservoirs from which they draw [4]. The first is called a dry steam plant, which produces electricity by drawing on dry steam from the earth’s core. The second is called a flash steam plant and a third binary cycle plant. Both flash steam and binary cycle plants use hot water reservoirs to generate electricity. While binary cycle plants can operate using water at temperatures between 107.2 °C and 182.2 °C, flash steam plants can operate at water temperatures higher than 182.2 °C [4]. As stated earlier, geothermal energy can also be used for heating and cooling applications. The first occurs by using geothermal heat pumps where the stable temperature of shallow ground (between 10 °C and 21.1 °C), is used for heating buildings in summer and cooling buildings in winter [4]. Geothermal energy can also be used as a “direct use” energy source [4]. In this system, the hot water from the water reservoir is passed through a heat exchanger, which also has a working fluid. The working fluid is kept separate from the hot water but is heated by the hot water. The working fluid is then passed through pipes and is used directly as a heat source. The cooled water is often disposed of in a disposal system. Direct use of geothermal energy is primarily made to heat buildings and it can also be used for agricultural purposes. The heating can be done on a large scale and supply many buildings, which is why it is often termed district heating. Direct use systems are also applied to melt snow on roads and sidewalks [4] (Fig. 1.8).

Fig. 1.8 Worldwide electricity generation using geothermal energy (2000–2018) [5]

10

1.1.9

1 Power Systems and Renewable Energy Systems

Ocean Energy

The ocean can be used to produce electricity in four principal ways [7], namely ocean thermal energy, tidal energy, wave energy and salinity gradient energy. Ocean thermal energy is produced from the sun’s heat; the surface water of the ocean is heated and used to turn a turbine, which in turn generates electricity. Tidal energy uses a dam to move water through turbines, which is then used to generate electricity. Wave energy is the utilization of energy in waves to propel a generator via a turbine or using the height difference in waves to propel a turbine. Salinity gradient energy is electricity generation based on the difference in salt concentration between two bodies of water. Pressure-retarded osmosis processes utilize this difference to generate electricity. It is worth noting that ocean energy technology is not as advanced as other technologies and hardly finds deployment (Fig. 1.9). Hydrogen Hydrogen is one of the universe’s most abundant elements, but is not a naturally occurring gas [4]. This therefore means hydrogen can only be produced by splitting it from compounds. There are several methods of producing hydrogen, including the electrolysis of water, gasification, and reformation of natural gas or the partial oxidation of methane. The most common method of producing hydrogen is known as reforming [4]. This entails the use of heat to split hydrogen molecules from carbon. This splitting process requires energy and leads to emissions of CO2. If this process is powered by RES without emissions, the hydrogen is referred to as CO2 neutral hydrogen or green hydrogen. Hydrogen is a key ingredient in the operation of fuel cells. Fuel cells, via redox reactions, convert the energy from hydrogen into electricity using an oxidizing agent (oxygen). Unlike conventional batteries, fuel cells need a constant stream of

Fig. 1.9 Worldwide electricity generation using ocean/marine energy (2000–2018) [5]

1.1 Introduction and Overview

11

Fig. 1.10 Configuration of RES integrated into a microgrid [8]

fuel (hydrogen) and oxygen to sustain chemical reactions and by extension operations. They can thus produce electricity continuously, as long as these two elements are in continuous supply. Hydrogen fuel cells are used in both low-power and high-power applications, among others in automobiles, transportation, and food processing [4]. Figure 1.10 below shows an example of a microgrid with a number of RES as the power source and battery storage.

1.2

Smart Grids

A smart grid means different things to different professionals and organizations; therefore, it is difficult to agree on a common or general technical definition. What can, however, be agreed upon is that a smart grid leads to the operational improvement and reliability of the power grid. The following are definitions of smart grids from various sources: The Energy Independence and Security Act of 2007 (EISA 2007) defines a smart grid as [9] the modernization of the Nation’s electricity transmission and distribution system to maintain a reliable and secure electricity infrastructure that can meet future demand growth.

The International Energy Agency (IEA) defines a smart grid as [10] an electricity network that uses digital and other advanced technologies to monitor and manage the transport of electricity from all generation sources to meet the varying

12

1 Power Systems and Renewable Energy Systems electricity demands of end users. Smart grids co-ordinate the needs and capabilities of all generators, grid operators, end users and electricity market stakeholders to operate all parts of the system as efficiently as possible, minimising costs and environmental impacts while maximising system reliability, resilience and stability.

The International Electro-technical Commission (IEC) defines a smart grid as [11] an electricity network that can intelligently integrate the actions of all users connected to it – generators, consumers and those that do both – in order to efficiently deliver sustainable, economic and secure electricity supplies. A smart grid employs innovative products and services together with intelligent monitoring, control, communication, and self-healing technologies to: facilitate the connection and operation of generators of all sizes and technologies; allow consumers to play a part in optimizing the operation of the system; provide consumers with greater information and choice of supply; significantly reduce the environmental impact of the whole electricity supply system; deliver enhanced levels of reliability and security of supply.

The United States Office of Electricity Delivery and Energy Reliability [12] defines a smart grid as a. class of technology that people are using to bring utility electricity delivery systems into the twenty-first century, using computer-based remote control and automation. These systems are made possible by two-way communication technology and computer processing that has been used for decades in other industries. They are beginning to be used on electricity networks, from the power plants and wind farms all the way to the consumers of electricity in homes and businesses. They offer many benefits to utilities and consumers – mostly seen in big improvements in energy efficiency on the electricity grid and in the energy users’ homes and offices.

Furthermore, the Energy Independence and Security Act of 2007 details ten distinguishing features of a smart grid [9], reproduced verbatim: (1) Increased use of digital information and control technology to improve reliability, security, and efficiency of the electric grid; (2) Dynamic optimization of grid operations and resources with full cybersecurity; (3) Deployment and integration of distributed resources and generation, including renewable resources; (4) Development and incorporation of demand response (DR), demand-side resources, and energy efficiency resources; (5) Deployment of “smart” technologies (real-time, automated, interactive technologies that optimize the physical operation of appliances and consumer devices) for metering, communications concerning grid operations and status, and distribution automation; (6) Integration of “smart” appliances and consumer devices; (7) Deployment and integration of advanced electricity storage and peak-shaving technologies, including plug-in electric and hybrid electric vehicles as well as thermal-storage air conditioning; (8) Providing consumers with timely information and control options; (9) Development of standards for communication and interoperability of appliances and equipment connected to the electric grid, including the infrastructure serving the grid;

1.2 Smart Grids

13

(10) Identification and lowering of unreasonable or unnecessary barriers to adoption of smart grid technologies, practices, and services. The core philosophy behind a smart grid is to transmit energy seamlessly from generators (both conventional and renewable powered generators) to end users in a cost-effective, reliable manner with advanced and intelligent infrastructure that permits the monitoring and control of network components and power flow. It is worth noting that in the context of a smart grid, the end users are not referred to solely as consumers but as prosumers [13, 14] because they simultaneously produce and consume electrical power. Monitoring and control of the power system components should be done in real time with the possibility of prompt remedial action in the event of blackouts or brownouts. The smart grid is also contingent on the implementation of advanced information communication infrastructure and digital technologies for effective condition monitoring and enhanced reliability. These technologies are applied at different spectrums of the power grids. Examples of such technologies include smart meters, flexible alternating current transmission system devices (FACTS) devices, vehicle-to-grid technologies, supervisory control and data acquisition, grid-to-vehicle technologies, and phasor measurement units (PMU) among others. The net result of these technologies is near real-time power system observation and control and the amplified incorporation of DR programs. DR programs seek to incentivize electricity consumers to curtail their electricity demand in response to price signals and customer infrastructure required [15, 16]. Other key components of the rapidly expanding smart grid include various energy and heat storage devices, the electrification of the transport system, prosumers, electric vehicles and RES [17]. Figure 1.11 shows a snapshot of the smart grid.

Fig. 1.11 Overview of a smart grid [18]

14

1.3

1 Power Systems and Renewable Energy Systems

Electricity Deregulation

The electricity industry is broadly broken down, based on its functions, into generation, transmission, and distribution. Generation is the creation of electrical power or electricity that occurs via the conversion of mechanical energy to electrical energy. The electric power obtained from generating plants is then transferred to distribution points in the power system. The distribution system is the last portion of the electric power supply process. Here electric power is transferred from the transmission system to the end users. Electric utilities use various mechanisms to perform these three core functions. The traditional mechanism adopted by electric utilities is referred to as vertical integration. Vertical integration occurs when a utility entity is solely in charge of generation, transmission, and distribution. Vertical integration has a host of advantages in theory, chief of which is the fact that having one entity control the whole spectrum of the power grid will lead to increased efficiency and minimal operating costs. A single entity controlling the whole spectrum of the grid from generation to distribution means that it could make the required capital-intensive investments in the electrical grid infrastructure and be poised to recoup investments with fair pricing. This pricing strategy for electric power is known as cost-based pricing and has been used by electric utility monopolies across the world. Vertical integration led to the birth of electricity utility monopolies. At the beginning of the electricity industry in the United States (prior to 1930), these monopolies were unregulated and were referred to as holding companies [19]. Holding companies were business entities formed by financiers and served as a vessel for a diverse range of business. Electricity production was just a business within the package and was generally profitable. Holding companies use the income from the sale of electricity to finance relatively riskier business ventures in the grouping. This led to a scenario where these holding companies were financially exposed, as they had little or no assets, only huge financial obligations and massive debts. The Great Depression had a massive impact on the electricity industry. At that time, holding companies with their business structure accounted for a large portion of the United States’ electricity production. When banks needed money and sought to recover this from holding companies, these companies folded and left many of their investors bankrupt. This catastrophic event led to the enactment of the Public Utilities Holding Companies Act of 1935 [19], which sought to regulate holding companies and prevent another collapse of the sector. It achieved this by prohibiting holding companies from consisting of utilities and other businesses. The Act also restricted the amount of debt holding companies could accrue and the geographical areas these utilities could serve. At the same time, Federal organs of state (such as the Security and Exchange Commission and the Federal Power Commission, now Federal Energy Regulatory Commission) were charged with powers to regulate interstate electricity transmission, while states had the prerequisite power to regulate electricity transmission within their state borders [19].

1.3 Electricity Deregulation

15

States could also determine and set the electricity retail price in their domains (Brennan et al. 1996), which factored in the utility’s cost of power production and marginal profit (cost-based pricing). If the utilities wanted a price increase, they had to show that there was a corresponding price increase in production costs and this had to be approved by the relevant regulator. However, advances in generation technologies led to a mounting push to introduce competition in the generating spectrum of the grid. The rationale for this is that competition in this segment will propel firms to utilize optimal energy-producing technologies at minimal cost. This push was also assisted by the drive to do away with fossil fuel-powered generator sources with their attendant environmental implications. Thus, industry players were being put under pressure to introduce a market at the generating spectrum of the grid. There was also a drive for introducing competition at the distribution spectrum of the grid. It was believed that having firms compete for customers at this spectrum of the grid would lead to the consumer being able to choose the cheapest source of electricity. This was to combat the ability of electric utilities (obtainable from vertical integration) to fix electricity prices arbitrarily; such tariffs were then passed on to the electricity consumers. Competition was envisaged to create efficiency in the distribution of electricity. The transmission spectrum of the grid historically was spared the drive for competition. This was in acknowledgement of the fact that there were (national) security implications of creating a market out of this spectrum. The introduction of competition in the generation and demand spectrum of the power grid led to the incorporation of market pricing in the pricing of electricity instead of cost-based pricing. Market pricing meant electricity was priced based on the interplay between forces of demand and supply. It is noteworthy that deregulation of the electricity industry was adopted for various reasons depending on whether it happened in a developed nation or a developing nation. In developed nations, deregulation was spurred on by the conventional reasons of price reduction, a drive for efficiency and minimizing the involvement of government. In developing nations/economies, electricity sector deregulation was motivated by the need to attract investors in the electricity sector in order to improve power system operations and reliability. However, the overarching aim (in either developing or developed nations) of deregulation was to improve efficiency and minimize the price of electricity, though this meant different things in different contexts. This ambiguity has also made it difficult to assess different deregulation market structures and the degree of their success uniformly [20]. The drive for deregulation of the electricity industry has also not had a uniform start in various jurisdictions. In the 1980s Chile led the movement with the enactment of the 1982 Electricity Act. England and Wales enacted the Electricity Act in 1989. Norway adopted its new Energy Act in 1990 and Colombia embraced deregulation in 1995, whereas Nigeria enacted the Electric Power Sector Reform Act in 2005. Various European nations followed the lead of Norway and by 2000, the European Union had mandated deregulation among its member nations [19]. In the United States, deregulation picked up steam in the 1980s in response to increasing electricity prices. By 1992, The Energy Policy Act had been enacted and

16

1 Power Systems and Renewable Energy Systems

paved the way for the deregulation of the electricity industry by states. California and Pennsylvania were among the first states to adopt deregulation and move from cost-based pricing to market pricing. Moving forward, 23 states had enacted laws to advance electricity deregulation by 2000, although the electricity crisis in California in that year dampened enthusiasm for deregulation. By 2014, seven states had abandoned their drive for deregulation, 16 states had active electricity markets and the remainder did not participate in deregulation [19] (Table 1.1). Table 1.1 Overview of electricity deregulation in organization for economic cooperation and development countries [20] Country

Liberalization

Electricity market

Consumption (TWh), 2002

Australia

Electricity Industry Act for Victoria (1994)

196

Austria

Law of Electricity Supply (1998)

Belgium

Law for the Organization of the Electricity Market, 29 April 1999 Alberta—Electric Utilities Act (2001), Ontario—Energy Competition Act (1998) The Energy Act (2000)

National Electricity Market (1997), Victoria Pool (1994) Energy Exchange Austria (2002) None

Canada

Czech Republic Denmark

Amendment to Danish Supply Act (1996, implemented 1998)

Finland

Electricity Market Act (1995)

France Germany

Law No. 2000–108 (2000) Act on the Supply of Electricity and Gas (1998) Electricity Law, 21 December, 1999 Electric Power Act (2001) Electricity Act (2003) Electricity Regulation Act (1999)

Greece Hungary Iceland Ireland Italy Japan Korea

Bersani Decree (1999) Amendments to Electric Utility Law (1995) Act on Promotion of Restructuring of the Electric Power Industry (2000)

55 79

Alberta Pool (1996) Ontario Market (2002)

487

Opera´tor trhu s elektrinou (2002) Nord Pool (1999 East Denmark; 2000 West Denmark) Finnish Electricity Exchange (1995) Powernext (2001) European Electricity Exchange (2000) None

55

None None Trading and Settlement Market (2000) Electricity Market (2004) None Korea Power Exchange (2001)

32

79 415 513 47 36 8 22 294 971 267

(continued)

1.4 Operation and Control

17

Table 1.1 (continued) Country

Liberalization

Electricity market

Consumption (TWh), 2002

Luxembourg

None—most power imported from neighbors

6

None

190

Netherlands

Law of 24 July 2000 on the organization of the electricity market Independent power producers allowed, no further liberalization The Electricity Act (1998)

101

New Zealand Norway

Energy Act and Companies Act (1992) Energy Act (1990)

Poland

Energy Act 1997

Portugal

Decree Laws 182/95, 183/95, 184/95, and 185/95 of 27 July 1995 Law on Energy (1998) Electricity Act (1994)

Amsterdam Power Exchange (1999) Electricity Market Company (1996) Norwegian Power Pool (1991), Nord Pool (1996) Polish Energy Exchange (2000) Iberian electricity market (2004)

Mexico

Slovakia Spain Sweden Switzerland Turkey United Kingdom United States

1.4

Law for the Supply of Electricity 10/95 Act defeated in 2002 referendum Electricity Market Law (2001) Electricity Act (1989),

PURPA (1978), Energy Policy Act (1992)

OMEL (1997), Iberian electricity market (2004) Nordpool (1996) None Electricity Pool of England and Wales (1990), NETA (2001) PJM (1998), ISO-NE (1999), NewYork (1999), ERCOT (2002)

36 107 117 42

24 218 138 55 118 344

3660

Operation and Control

The operation and control of a power system is concerned with the optimal management of the power system inclusive of its power sources, transmission lines and load. This very difficult task requires highly skilled personnel, operations, and algorithms. The North American Electric Reliability Council has delineated key areas for optimal operation and control of power systems [1]:

18

1 Power Systems and Renewable Energy Systems

i. Power generation and demand must be balanced. ii. Reactive power supply and demand must be balanced to maintain acceptable voltage levels. iii. Transmission line power flows must be monitored constantly to avoid line limits violations. iv. It must be ensured that power system stability and reliability are maintained. v. Preparation for emergencies must be such that reliability is maintained even if an important component fails. This includes conventional power system operational and control issues such as power flow, optimal power flow (OPF), unit commitment, economic dispatch, generator maintenance scheduling (GMS), generator and transmission expansion planning, FACTS devices placement, and PMU placement, etc. These issues have various time scales ranging from seconds (optimal power flow) to weeks (generator maintenance scheduling) and years (generator and transmission expansion planning). Mathematical optimization can be a useful tool in achieving optimal operation and control of the power system for the aforementioned power system issues. A mathematical model typically maximizes or minimizes power system parameters. This can be the minimization of generator fuel costs, the maximization of output of power generators, minimization of generator emissions, and minimization of operating costs, etc. Recent advances in current power systems have introduced modern operational issues such as the grid integration of renewables, DR, prosumer integration, and electric vehicles, etc. The advent of the Fourth Industrial Revolution and its associated technologies such as the internet of things (IoT), blockchain, machine learning, big data etc. have also been deployed to increased grid reliability and stability. IoT and blockchain technologies have been used for smart home management systems and have led to more efficient operation and control of the power system. Classification, clustering, and regression have been used for solar radiation modeling, wind speed modeling, price forecasting, and load forecasting, etc. [21].

1.5

Summary

This chapter introduced a power system and detailed the various sectors in power system operation. Furthermore, recent RES that have increasingly been adopted were discussed. The chapter also examined various definitions of a smart grid and what constitutes a smart grid. The deregulation of the electric power industry across various climes was mentioned and the chapter was concluded with an introduction to operational and control issues in a power system.

References

19

References 1. U.S.-Canada Power System Outage Task Force (April 2004) “Final report on the August 14, 2003 blackout in the United States and Canada: causes and recommendations” 2. IEA, “Data and Statistics—IEA” (2020) [Online]. Available: https://www.iea.org/data-andstatistics?country=WORLD&fuel=Energy%20supply&indicator=Electricity%20generation% 20by%20source. [Accessed: 09 Jul 2020] 3. IEA, “Data and Statistics—IEA” (2020) [Online]. Available: https://www.iea.org/data-andstatistics?country=WORLD&fuel=Energy%20consumption&indicator=Electricity% 20consumption. [Accessed: 09 Jul 2020] 4. Tromly K (2001) “Renewable energy: an overview,” National renewable energy, laboratory, golden, CO (US) 5. International Renewable Energy Agency (2020) “Trends in Renewable Energy” [Online]. Available: https://public.tableau.com/views/IRENARETimeSeries/Charts?:embed=y&: showVizHome=no&publish=yes&:toolbar=no. [Accessed: 09 Jul 2020]. 6. Blanco MI (2009) The economics of wind energy. Renew Sustain Energ Rev 13(6–7):1372– 1382 7. García Vera Y, Dufo-López R, Bernal-Agustín J (2019) Energy management in microgrids with renewable energy sources: a literature review. Appl Sci 9(18):3854 8. Irena.org (2020) “Ocean energy” [Online]. Available: https://www.irena.org/ocean. [Accessed: 09 Jul 2020] 9. Energy Independence and Security Act of 2007 10. IEA (2011) Technology roadmap—smart grids. Paris: IEA. [Online]. Available: https://www. iea.org/reports/technology-roadmap-smart-grids. [Accessed: 09 Jul 2020] 11. Iec.ch (2020) “IEC—Smart grid > Background—What is Smart Grid?” [Online]. Available: https://www.iec.ch/smartgrid/background/explained.htm. [Accessed: 09 Jul 2020] 12. Energy.gov (2020) “Grid modernization and the smart grid” [Online]. Available: https://www. energy.gov/oe/activities/technology-development/grid-modernization-and-smart-grid. [Accessed: 09 Jul 2020] 13. Damisa U, Nwulu NI, Sun Y (2018) A robust energy and reserve dispatch model for prosumer microgrids incorporating demand response aggregators. 10(5) 14. Damisa U, Nwulu NI, Sun Y (2018) Microgrid energy and reserve management incorporating prosumer behind-the-meter resources. IET Renew Power Gener 12(8):910–919 15. Fahrioglu M, Nwulu NI (2012) Investigating a ranking of loads in avoiding potential power system outages. J Electr Rev (Przeglad Elektrotechniczny) 88(11a):239–242 16. Nwulu NI, Fahrioglu M (8–11 May 2011) “A neural network model for optimal demand management contract design”, Proceedings of the 10th international conference on environmental and electrical engineering (EEEIC2011), Rome, Italy 17. Gbadamosi S, Nwulu NI, Sun Y (2018) Multi-objective optimization for composite generation and transmission expansion planning considering offshore wind power and feed in tariffs. IET Renew Power Gener 12(14):1687–1697 18. IEA (2014) Energy technology perspectives 2014, IEA, Paris https://www.iea.org/reports/ energy-technology-perspectives-2014. [Accessed: 09 Jul 2020] 19. Owens S (2017) “Measuring the effect of electric utility deregulation on residential retail prices in a Midwestern state”. MPA/MPP Capstone projects, University of Kentucky 20. Alsunaidy A, Green R (2006) Electricity deregulation in OECD (organization for economic cooperation and development) countries. Energy 31(6–7):769–787 21. Nwulu NI, Agboola PO (18–25 May 2012) “Modelling and predicting electricity consumption using artificial neural networks”, Proceedings of the 11th international conference on environmental and electrical engineering (EEEIC2012), Venice, Italy 22. Unece.org (2020) [Online]. Available: https://www.unece.org/fileadmin/DAM/energy/se/ pdfs/eneff/eneff_h.news/Smart.Grids.Overview.pdf. [Accessed: 09 Jul 2020]

Chapter 2

Demand Side Management

2.1

Introduction

The advancement in the power system and its rapid modernization is often referred to as the smart grid. Today’s smart grid is grappling with increased demand due to a rapidly increasing population and increased demand for operational reliability. Power system regulators and operators have two principal ways of dealing with these challenges. The primary method is increasing the capacity of the power system, from the generation spectrum through the transmission spectrum to the distribution spectrum. The problem with this approach is that it often involves a huge financial outlay to which the utility and consumers might not be favorably disposed, as these expenses are usually passed on to the consumers. An allied reason is the fact that there are often environmental implications to expanding the capacity of the power system. It is widely known that power system plants, transmission lines and infrastructure generally leave a huge environmental footprint and depending on some technologies, might contribute to increased emissions and global warming. The second approach, which is favorably regarded by power system operators and regulators, involves DSM programs. DSM programs are conceptualized to encourage consumers to be participants in the power system by curtailing their power consumption through a host of programs, behavioral changes and technological inventions. One of the earliest advocates for DSM programs was Amory Lovins [1] who posited that “A kilowatt-hour saved is just like a kilowatt-hour generated … so they should be treated alike.” Amory Lovins, in “Saving Gigabucks with Negawatts” [2], defined a negawatt as the unit of electricity saved and the inverse of the watt, which is the unit of energy consumed. In essence, the aim of DSM programs is to alter a consumer’s electricity consumption pattern actively and reduce energy consumption. This aim can be attained by peak clipping, valley filling, load shifting, strategic conservation, strategic load © Springer Nature Switzerland AG 2021 N. Nwulu and S. L. Gbadamosi, Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language, Green Energy and Technology, https://doi.org/10.1007/978-3-030-00395-1_2

21

22

2 Demand Side Management

Fig. 2.1 An overview of DSM, energy efficiency and demand response programs [3]

growth and flexible load shape [3]. DSM programs can be applied as either short-term or long-term strategic interventions that can be deployed in times of power system strain and stress. Furthermore, they can be deployed either throughout the power system or at previously identified problem points. DSM can be categorized into two, namely energy efficiency programs and demand response programs. Both programs are briefly detailed in the next subsections. Figure 2.1 presents a detailed overview of DSM programs.

2.2

Energy Efficiency Programs

Energy efficiency programs seek to curtail consumers’ electricity consumption while maintaining the consumers’ operational abilities. These programs are conceptualized to reduce the energy consumed by buildings, organizations or communities without altering the activities for which the consumers use the energy. Energy efficiency programs are independent of the location of the end users and the time profile of energy consumption. An example is replacing incandescent bulbs with compact fluorescent tubes, which will reduce the energy consumed by the user independent of the location of the user and the lighting time profile of the user. Energy efficiency programs are a significant cost-saving measure for consumers and utility alike and lead to a significant reduction in environmental emissions. Energy efficiency programs/investments also yield return on investment owing to the cost

2.2 Energy Efficiency Programs

23

savings the consumer or utility accrues. To contextualize or explain energy efficiency programs properly, we will detail an example of an energy efficiency program below.

2.2.1

Building Retrofit Energy Efficiency Programs

When building retrofit programs are undertaken, consumers replace their inefficient energy devices with energy-efficient ones. These programs are a classical straightforward kind of energy efficiency program and have the potential to save costs (especially for large buildings and other structures) and to minimize emissions. To provide more insight into the structure and objectives of energy efficiency programs, we will detail a building retrofit program in South Africa, called the Standard Product Program [4]. The Standard Product program was designed by Eskom, the South African utility monopoly, to encourage various kinds of customers (commercial, industrial and agricultural customers) to discard their existing energy-inefficient devices or technologies and substitute those with energy-efficient fixtures or technologies. These replacement technologies or fixtures can range from a single component to an energy-saving system and may even be lighting fixtures, energy-efficient showerheads or heat pumps etc. Eskom defined a framework to guide the choice of technologies to be contained within the Standard Product Program, which are that the technology should be [4]: • A specific ‘off-the-shelf’ product, e.g. a lamp, fitting, heat pump, etc. satisfying a specific requirement, which appeals to a wide market; • Approved by Eskom to be used as a replacement for less energy-efficient technologies (compact fluorescent lamps, incandescent bulbs, etc.); • A component, sub-assembly or complete “machine”; and/or • Programmable; it may even be a control system, provided a predicable saving can be attached to the technology. Eskom will give participating customers rebates, since the customers would have incurred costs when installing the energy-efficient technologies. Eskom has stipulated the following conditions: • • • •

Total demand reduction must be between 1 and 250 kW. The total energy savings should exceed 2 MWh/a. The maximum rebate/technology is restricted to ZAR 750 000. The customer should dispose of all inefficient technologies and foot the bills for the installation of the new technologies. • The customer should possess the appropriate tax clearance and their electricity account should not be in arrears.

24

2.3

2 Demand Side Management

Demand Response Programs

DR, one of the principal DSM techniques, can be simply defined as “a change in electric usage by end-use customers from their normal consumption patterns in response to changes in the price of electricity over time, or to incentive payments designed to induce lower electricity use at times of high wholesale market prices or when system reliability is jeopardized” [5, 6]. It follows that DR programs have the goal of reducing the consumer’s electricity load profile but uses either incentives or time-varying price signals in a bid to achieve this aim. DR programs have the added advantage of minimizing operating costs, minimizing emissions and enhancing the reliability and efficiency of power systems. These advantages hold true for both electricity markets and vertically integrated power systems. However, in electricity markets, DR programs lead to reduced wholesale market prices [7]. There are two broad-based classifications of DR programs. The first type of DR program is price-based demand response [6]. In price-based DR programs, there is a time variation in electricity price [6]. This means that the same quantity of electricity will be priced differently, contingent on the time of day. The pricing structure largely depends on the existing demand, as demand varies depending on the time of day. Thus, when demand for electricity is high, electricity is expensive and when demand is low, electricity is priced lower. This has the advantage of altering consumers’ behavior and causing them to use electricity at times when the price is lower. Examples of price-based DR programs include time-of-use rates (TOU), real-time pricing (RTP), critical peak pricing (CPP), extreme day pricing, and extreme day—critical peak pricing [6, 8]. The second type of DR program is IB-DR programs [9]. IB-DR programs, as the name implies, offer incentives to consumers who curtail their electrical load at times of power system pressure. The incentive can take the form of either lower electricity tariffs or monetary payments. IB-DR programs often include penalties in the event that consumers do not curtail their electricity at the required time and to the required degree. Examples of IB-DR include direct load control (DLC), interruptible services (IS), emergency demand response programs (EDRP), capacity market programs (CMP), demand-bidding/buyback programs and ancillary market services [9, 10]. Consumers are not compelled to participate in DLC and EDRP and there are no penalties. The two programs with penalties for defaulting consumers are IS and CMP. Figure 2.2 shows the different types of DR programs.

2.3.1

Price-Based Demand Response

Historically, tariffs for electricity consumers have been fixed and set in advance by utilities and regulators [11]. As the electricity industry experienced a transition from

2.3 Demand Response Programs

25

Fig. 2.2 The classification of DR programs [8]

vertical integration to a market, there was a corresponding drive for pricing schemes that mirrored the price variation of the power system, since power system operational costs vary daily, weekly or seasonally. The expectation is that customers would leverage the price variation and adjust their consumption to take advantage of cheaper prices [6, 10]. In the next subsections, we briefly detail the key price-based DR programs.

2.3.1.1

Time-Of-Use Pricing

TOU pricing is a price framework where electric power is priced differently based on time or seasonal variations [12]. Under the conventional TOU program, electricity is priced differently at peak periods and off-peak periods, depending on when electricity demand typically peaks. Some countries even have peak periods, off-peak periods and standard periods. Under a seasonal TOU pricing strategy, electric power is priced differently based on the seasons. Thus, seasons with higher demand for electricity, i.e. summer and winter, will have a different pricing scheme to seasons with lower demand, i.e. fall and spring. TOU programs are typically optional and have criteria for participating customers, with most consumers being industrial consumers.

2.3.1.2

Real-Time Pricing

RTP frameworks provide a pricing strategy where electricity is more aligned to (near) real-time market conditions as opposed to TOU pricing frameworks that are predetermined based on prior time/seasonal prices [12]. It is often difficult to obtain practical real-time prices, so in practice prices are given to consumers from as close as a day ahead to an hour ahead of real time [10]. These prices are based on short-term forecasts of power system operational and market conditions.

26

2.3.1.3

2 Demand Side Management

Critical Peak Pricing

In a bid to avert the hourly variation obtainable from RTP over a significant portion of operating hours, CPP is a pricing framework that attempts to model near “real”time market conditions only at specific shorter time intervals/operating hours [12]. Thus, RTP is only deployed at those hours when the power system is extremely highly challenged and this may range from a few hours to a few days in a year, depending on the robustness and reliability of the power system. Consumers that sign up for the CPP program will receive notice from the utility on when the power system is expected to be significantly challenged (this may range from a few hours to a day) and the consumer is expected to take corresponding action by either curtailing load demand or paying for the more expensive electricity.

2.3.1.4

Price Elasticity of Electrical Demand

Price elasticity is a metric used to measure the amount of change in consumer demand due to a price change. The conventional notion is that consumers’ demand for goods or services reduces as the price increases. This magnitude of this change is referred to as the elasticity co-efficient. The mathematical formula for elasticity is given below: E¼

Dd=d0 ; Dp=p0

ð2:1Þ

E¼

Ddp0 ; Dpd0

ð2:2Þ

where Dd, Dp, d0 , and p0 are the change in demand, change in price, initial demand, and initial price, respectively [6]. Equations 2.1 and 2.2 are termed the self-elasticities, as they measure the change in demand for specific goods/services based on a price change for those goods/services. If, on the other hand, one wants to obtain the change in commodity of goods/services based on price change (increase/ decrease) of another commodity, then this is referred to as cross-elasticities. The values for self-elasticities are usually −ve and +ve for cross-elasticities. Deploying this concept to electricity markets, if a price increase in peak periods therefore causes a change in consumer demand in peak periods, then we measure this with self-elasticities and if this same price increase in peak periods leads to a change in demand in off-peak periods, we measure this with cross-elasticities. The mathematical formula for cross-elasticities is given below: Eði; jÞ ¼

Ddi =d0i Dpj =p0j

ð2:3Þ

2.3 Demand Response Programs

27

Eði; jÞ ¼

Ddi p0j Dpj d0i

ð2:4Þ

where i and j represent differing time frames such as off-peak and peak, d0i is the original consumer demand during i, p0j is the initial electricity price during j, Dpj is the price variation during j and Ddi is the consumer demand variation during i. Assuming a daily scheduling interval with various peak and off-peak time periods, we use an elasticity matrix to represent the various changes in consumer demand to price changes (increase/decrease). The +ve off diagonal elements depict the cross-elasticities and the −ve diagonal elements depict the self-elasticities. Equation 2.5 gives the structure of an elasticity matrix for a given market over a 24-h time horizon. 2

Eð1; 1Þ 6 Eð2; 1Þ 6 E¼6 6 ... 4 ... Eð24; 1Þ

2.3.1.5

Eð1; 2Þ Eð2; 2Þ ... ... Eð24; 2Þ

... ... Eði; jÞ ... ...

... ... ... ... ...

3 Eð1; 24Þ Eð2; 24Þ 7 7 7 ... 7 5 ... Eð24; 24Þ

ð2:5Þ

Determining Price Elasticity for Time-Of-Use Demand Response Programs

Using the concept of price elasticity matrices, it is possible to determine the load economic profile for price-based DR programs. An example is shown below on how to use price elasticity matrices to determine the load economic profile for the TOU DR program[6]: We begin by defining Bðdi Þ as the total consumer benefit in time interval i that accrues from the utilization of di kWh of energy and pi as the electricity price during interval hour i. The consumers’ profit is denoted by: Si ¼ Bðdi Þ  pi di

ð2:6Þ

For consumers’ profit maximization, @Si =@di should be zero and we thus obtain: @Bðdi Þ ¼ pi @di

ð2:7Þ

If we utilize the quadratic benefit function (any other benefit function can be used), then the total consumer benefit can be represented thus [6]:   di  d0i Bðdi Þ ¼ Bðd0i Þ þ p0i ðdi  d0i Þ 1 þ 2Eði; iÞ:d0i

ð2:8Þ

28

2 Demand Side Management

where d0i p0i Eði; iÞ Bðd0i Þ

is is is is

the the the the

initial consumer demand during i, i = 1,2,…,24; initial electricity price during i, i = 1,2,…,24; self-elasticity; consumer benefit at d0i ;   @Bðdi Þ di  d0i ¼ p0i 1 þ : @di Eði; iÞ:d0i

ð2:9Þ

Equating (2.7) to (2.9), it is determined that:  di ¼ d0i

 Eði; iÞ½pi  p0i  1þ : p0i

ð2:10Þ

Assuming multi-period elastic loads with consumers being able to reschedule, the total demand at interval i is a function of electricity price variations during i = 1, i 2… T. Assuming further that the linearity assumption Dd Dpj is constant for i, j = 1, 2, 3,…0.24, the relationship between electricity demand and price for a 24-h TOU program is: 2 6 6 di ¼ d0i 61 þ 4

24 P j¼1

3 Eði; jÞ½pj  p0j 7 7 7: 5 p0j

ð2:11Þ

Combining Eq. 2.10 and 2.11 into one equation gives: 2

3 Eði; jÞ½p  p  j 0j 6 7 Eði; iÞ½pi  p0i  j¼1 6 7 di ¼ d0i 61 þ þ 7: 4 5 p0i p0j

2.3.1.6

24 P

ð2:12Þ

Appliance Scheduling Using Time-Of-Use Demand Response

In [6], a TOU DR framework is presented. There are three classes of DR loads: flexible loads, inflexible loads, and night-time loads. Inflexible loads are defined as those loads that affect the comfort of consumers and thus must always be switched on. Examples for residential consumers are space/water-heating technologies and perhaps cooking and refrigerating equipment. Some industrial loads are also inflexible, especially for critical industrial processes. Flexible loads are those loads for which customers can easily adjust their demand in response to price variations. For residential consumers, examples include

2.3 Demand Response Programs

29

electric food-processing equipment, etc. Nighttime loads are defined as off-peak leads, which are often time intervals with the lowest tariffs. For residential consumers, this could be washing machines and dryers. Each load class has three different price elasticity matrices for a 24-h scheduling interval. The matrices are 24  24 square matrices and the difference between the three is based on the position of the non-zero elements. Equations 2.13–2.15 show the three matrices. 3 Eð1; 1Þ Eð1; 2Þ 0 0 0 0 0 7 6 Eð2; 1Þ Eð2; 2Þ 0 0 0 0 0 7 6 7 6 0 0 Eð3; 3Þ Eð3; jÞ 0 0 0 7 6 7; 6 0 Eði; 3Þ Eði; jÞ Eði; 22Þ 0 0 ¼6 0 7 7 6 0 0 0 Eð22; jÞ Eð22; 22Þ 0 0 7 6 4 0 0 0 0 Eð23; 22Þ Eð23; 23Þ Eð23; 24Þ 5 0 0 0 0 0 Eð24; 23Þ Eð24; 24Þ 2

Einf

lexible

ð2:13Þ 2

Eflexible

Eð1; 1Þ 6 Eð2; 1Þ 6 6 Eð3; 1Þ 6 6 Eð4; 1Þ 6 6 Eð5; 1Þ 6 6 0 6 6 0 ¼6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 4 0 0

0 Eð2; 2Þ Eð3; 2Þ Eð4; 2Þ Eð5; 2Þ Eð6; 2Þ 0 0 0 0 0 0 0 0

0 0 Eð3; 3Þ Eð4; 3Þ Eð5; 3Þ Eð6; 3Þ Eð7; 3Þ 0 0 0 0 0 0 0

0 0 0 Eð4; jÞ Eð5; jÞ Eð6; jÞ Eð7; jÞ Eð8; jÞ 0 0 0 0 0 0

0 0 0 0 0 0 0 0 Eð19; 22Þ Eð20; 22Þ Eð21; 22Þ Eð22; 22Þ 0 0

0 0 0 0 0 0 0 0 Eð19; 23Þ Eð20; 23Þ Eð21; 23Þ Eð22; 23Þ Eð23; 23Þ 0

3 0 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7; 7 0 7 7 0 7 Eð20; 24Þ 7 7 Eð21; 24Þ 7 7 Eð22; 24Þ 7 7 Eð23; 24Þ 5 Eð24; 24Þ

ð2:14Þ 2

Enighttime

Eð1; 1Þ Eð1; 2Þ 6 Eð1; 2Þ: Eð2; 2Þ 6 6 0 0 6 0 0 ¼6 6 6 0 0 6 4 Eð1; 23Þ Eð2; 23Þ Eð1; 24Þ Eð2; 24Þ

3 ... Eð1; jÞ ... Eð1; 23Þ Eð1; 24Þ ... Eð2; jÞ ... Eð2; 23Þ Eð2; 24Þ 7 7 7 Eð3; 3Þ 0 0 0 0 7 7: 0 Eði; jÞ 0 0 0 7 7 0 0 Eð22; 22Þ 0 0 7 ... Eði; 23Þ ... Eð23; 23Þ Eð23; 24Þ 5 ... Eði; 24Þ ... Eð23; 24Þ Eð24; 24Þ

ð2:15Þ The above information can be used to obtain an optimal minimal cost schedule for inflexible, flexible and night-time loads under a TOU tariff plan [8]. This is referred to as appliance scheduling.

30

2.4

2 Demand Side Management

Incentive-Based Demand Response

When using IB-DR programs, electricity consumers shut down or curtail their electricity load and in return receive incentives for doing so. An overview of major IB-DR programs is detailed below:

2.4.1

Direct Load Control

DLC programs allow the utility supplier to leverage on advanced monitoring and control technologies to switch off or curtail the operations of specific customer equipment in a bid to provide relief to the power grid [10]. The customers are typically residential or smaller industrial or commercial organizations. In return, the consumers are given either a monetary rebate or a reduced electricity tariff as their compensation for participation. Typically, the consumers do not have control over their participation in DLC programs. There are, however, some DLC programs where consumers retain control over their equipment and can choose to decline to participate when given the signal to curtail energy consumption. In such a case, consumers are penalized depending on the permissible number of times they may choose to decline participation. DLC programs depend on excellent communication infrastructure for proper functioning. DLC is typically applied to energy-guzzling appliances such as water- and space-heating equipment.

2.4.2

Interruptible Programs

Interruptible programs are geared towards large-scale energy consumers’ (loads > = 1 MW) in order to incentivize them to discontinue their operations for a pre-determined period of time [12]. These large-scale energy consumers can be activated quickly in the event the power system becomes unexpectedly stressed or strained and either have their own back-up power-generation schemes or can shift their operational schedule to when the system is less stressed [12]. The incentive in these cases is rates, which are typically lower than the conventional tariffs, and these programs have strict terms, with severe penalties for non-compliance. The programs, for example, will explicitly spell out the terms of the interruption, quantity of power to be shut down by participating consumers, reduced rates, which are the incentives for compliance, and the penalties for non-compliance.

2.4 Incentive-Based Demand Response

2.4.3

31

Curtailable Load Program

The curtailable load program is essentially a variant of the interruptible program through which energy consumers are again incentivized to provide power system relief. However, unlike the interruptible program, these companies are not requested to discontinue their operations completely, but to reduce or curtail their energy consumption [12]. Another key difference from the interruptible program is that the consumers do not necessarily have to be large-scale consumers; the program often involves a mix of large-scale and medium-to-large-scale consumers. Yet another difference is that typically the penalties for curtailable load programs and the window frame for compliance are lower than those of the interruptible program. Furthermore, the nature of the program allows consumers to choose the devices/ loads to be reduced in order to achieve the previously agreed target for load reduction, as this is not enforced by the utility, unlike the DLC program.

2.4.4

Demand-Bidding Programs

In demand-bidding programs, consumers are incentivized to curtail their electricity load demand [12]. Two formats are used for this program. The first is the customer-bidding program and the second is termed sponsor bidding. In the customer-bidding program, consumers are the active bidders and submit a day-ahead bid to the utility, indicating the amount of load they are willing to curtail and at what price. The utility then considers the bids received in addition to the forecast power system condition and decides on the bids to be accepted [12]. In the sponsored bidding program, the utility is the active participant and announces a predetermined incentive (price) it is willing to pay for demand curtailment. Electricity consumers then consider the amount and in return indicate the quantum of load they are willing to curtail, based on the utility’s announced price. Under both programs, the consumer is thereafter paid upon fulfillment of its own side of the deal [12].

2.4.5

Game Theory Demand Response Programs

A critical issue determined in IB-DR programs is the amount/quantity of the incentive offered by the utility. If the incentive is too little, consumers will not be interested in participating [13–15]. On the other hand, the utility cannot afford to give excessively high incentives because of budget and affordability issues. The utility therefore has to offer incentives that will truly attract consumer participation, which means that the incentive should ideally be greater than the cost of power interruption. To combat this challenge, game theory has been deployed to design optimal DR programs.

32

2 Demand Side Management

In references [13, 14], the authors delineate three key criteria that incentive-based demand management programs should consider: • IB-DR programs should distinguish between different customers. Different customers imply dissimilar loads and dissimilar load outage costs. • In order to offer attractive consumer incentives, the utility should be able to determine in monetary terms ballpark ranges of what these consumer load outage costs amount to. • Incentive-based programs also have to factor in the locations of the consumers in calculating incentives, as different consumers have different impacts on the power grid. An optimal approach to designing IB-DR programs while incorporating all three design criteria is detailed below.

2.4.5.1

Game Theory Based Demand Response Formulations

We begin by defining key parameters: • h differentiates different kinds of customers and is normalized between 0 and 1 (0  h  1) with h = 1 depicting the consumer most inclined to shed power and h = 0 depicting the consumer least inclined to shed electric power. • x is the amount of power either curtailed or shut down by an electricity consumer. • c(x, h) is customer h incurred cost due to load reduction/curtailment of x MW • k factors in the locational impact different consumers have on the grid and depicts in monetary terms the cost of failure to supply power to particular consumers based on their location on the grid. In the event of the grid being strained, it might be better for the utility not to supply certain locations/buses and the utility can determine the value typically from an optimal power flow (OPF) calculation. This is also referred to as the “value of power interruptibility” (k) [13, 14]. We can give the benefit function of the consumer as: V1 ðh; x; yÞ ¼ y  cðh; xÞ

ð2:16Þ

where y is the incentive received by the consumer for shedding load. In order to ensure attractiveness to the consumer, V1  0. In the same vein, the utility’s benefit function is given as: V2 ðh; kÞ ¼ kx  y

ð2:17Þ

2.4 Incentive-Based Demand Response

33

The utility’s aim will therefore be to maximize its benefit: max½kx  y:

ð2:18Þ

x;y

The assumed form of the consumers cost function is given below: cðh; xÞ ¼ K1 x2 þ K2 x  K2 xh

ð2:19Þ

where K1 and K2 are the cost function coefficients. Equation 2.19 is a quadratic cost function. The cubic and exponential cost functions are other kinds of cost functions also used [13]. However, cost functions must satisfy the following criteria [13, 14]: • • • •

An increase of h decreases marginal cost and increases marginal benefit. The marginal cost should be non-negative. The cost function should be convex (increasing marginal cost). If a customer does not reduce use or curtail any power, the cost should be zero ( c( h, 0) = 0). • cðh; xÞ ¼ K1 x2 þ K2 x  K2 xh (quadratic cost function). • The K2 xh term sorts customers by way of h. • @c @x ¼ 2K1 x þ K2  K2 h.

Assuming there are J customers and one utility, yj is the monetary value of payments to customer j with the utility benefit for customer j uj ¼ yj  ðK1 x2j þ K2 xj  K2 xj hj Þ; for

j¼ 1;. . .;J:

ð2:20Þ

Therefore, the total utility benefit for J customers is: u0 ¼

J X

kj x j  y j

ð2:21Þ

j¼1

The utility will want to maximize this benefit. max x;y

J  X

kj x j  y j



ð2:22Þ

j¼1

The utility will also want to enforce two requirements, termed the “individual rationality constraint” and the “incentive compatibility constraint”, both of which are represented below: yj  ðK1 x2j þ K2 xj  K2 xj hj Þ  0; for

j¼ 1;. . .;J:

ð2:23Þ

34

2 Demand Side Management

yj  ðK1 x2j þ K2 xj  K2 xj hj Þ  yj1  ðK1 x2j1 þ K2 xj1  K2 xj1 hj1 Þ; for ð2:24Þ j¼ 1;. . .;J: The “individual rationality constraint” seeks to enforce the requirement that each consumer has a +ve benefit while the “incentive compatibility constraint” enforces compensation corresponding to the amount of power shed.

2.5

Summary

In this chapter, DSM programs encompassing energy efficiency and DR programs were detailed. DR programs are subdivided into incentive-based and price-based programs, which were considered in detail. Examples of energy efficiency programs (a retrofit program), price-based DR (TOU appliance scheduling) and IB-DR programs (game theory-based incentive-based DR) were discussed in detail and their mathematical formulations were presented.

References 1. Wirl F (1997) The economics of conservation programs. Kluwer Academic Publishers, Dordrecht, The Netherlands 2. Lovins A (1985) Saving gigabucks with negawatts. Public Utilities Fortnightly 115(6):19–26 3. North American Electric Reliability Corporation (2011) Demand response availability report, March 2013 4. Nwulu N, Sun Y “Optimal incentivized demand response and building retrofits”, 3rd International conference on advances in computing, communication and engineering, ICACCE 2016, Durban, South Africa 28, 29 November 2016 5. Department of Energy (Feb 2006) U. S. benefits of demand response in electricity markets and recommendations for achieving them 6. Nwulu NI, Xia X (2015) A combined dynamic economic emission dispatch and time of use demand response mathematical modelling framework. J Renew Sustain Energ 7(4) 7. Nwulu NI, Fahrioglu M (2011) Power system demand management contract design: a comparison between game theory and artificial neural networks. Int Rev Model Simul 4 (1):104–112 8. Nwulu NI, Fahrioglu M (8–11 May 2011) “A neural network model for optimal demand management contract design”, Proceedings of the 10th international conference on environmental and electrical engineering (EEEIC2011). Rome, Italy 9. Nwulu NI, Xia X (2015) Multi-objective dynamic economic emission dispatch of electric power generation integrated with game theory based demand response programs. Energ Convers Manage 89(1):963–974 10. National Action Plan for Energy Efficiency (2010) Coordination of energy efficiency and demand response. Prepared by Charles Goldman (Lawrence Berkeley National Laboratory), Michael Reid (E Source), Roger Levy, and Alison Silverstein 11. Fahrioglu M, Nwulu NI (Nov 2012) Investigating a ranking of loads in avoiding potential power system outages. J Electr Rev (Przeglad Elektrotechniczny), Warsaw, Poland 88(11a): 239–242

References

35

12. Rocky Mountain Institutie (30 April 2006) Demand response: an introduction, overview of programs, technologies and lessons learned. Southwest Energy Efficiency Project 13. Fahrioglu M, Alvarado FL (May 2001) Using utility information to calibrate customer demand management behavior models. IEEE Trans Power Syst 16(2) 14. Fahrioglu M, Alvarado FL (Nov 2000) Designing incentive compatible contracts for effective demand management. IEEE Trans Power Syst 15(4) 15. Damisa U, Nwulu NI, Sun Y (2018) Microgrid energy and reserve management incorporating prosumer behind the-meter Resources. IET Renew Power Gener 12(8):910–919

Chapter 3

Mathematical Optimization Modeling and Solution Approaches

3.1

Introduction

Mathematical optimization is a branch of applied mathematics that is concerned with determining the best option among a collection of options with respect to a clearly defined objective or yardstick. The word optimization is derived from the word optimal, which can loosely be translated to mean best, peak, prime. Thus, mathematical optimization is mathematically determining the best option or alternative from a number of choices. Mathematical optimization modeling has found application in diverse fields, subject matters, and industries. It is widely used in engineering, supply chain modeling, finance, and medicine; in fact, in just about any field of human endeavor. A mathematical optimization model consists of three critical elements. The first is a well-defined objective, which the modeler wants either to maximize or minimize. In the terminology of mathematical optimization, this is referred to as the objective function. The second is variables, which are elements that affect one’s objective and that one is able to vary. The last critical element are constraints, which are mathematical equations that impose relevant limits on one’s variables. All three, objectives, variables, and constraints, can be abstracted from any problem of interest in any domain to find the “optimal” or “best” option among a set of competing alternatives. This abstraction process is referred to as mathematical modeling and it has become a scientific endeavor owing to the need to deploy mathematical optimization in very complicated and complex domains. The resulting set of equations (consisting of objectives, variables, and constraints) is known as a mathematical model. A mathematical model is typically written in the general form: Minimize or Maximize FðxÞ

© Springer Nature Switzerland AG 2021 N. Nwulu and S. L. Gbadamosi, Optimal Operation and Control of Power Systems Using an Algebraic Modelling Language, Green Energy and Technology, https://doi.org/10.1007/978-3-030-00395-1_3

ð3:1Þ

37

38

3 Mathematical Optimization Modeling and Solution Approaches

Subject to gðxÞ\ ¼ b hð xÞ ¼ c

ð3:2Þ ð3:3Þ

Equation (3.1) is the objective function, Eq. 3.2 is the inequality constraint and Eq. 3.3 is the equality constraint. Mathematical optimization uses computing machinery to solve the resulting model and requires a solver, which simply put is a mechanism for reading the mathematical model that provides a practical solution. There are various classifications of mathematical optimization models. A common classification is based on the presence of constraints. Thus, optimization models with constraints are referred to as constrained optimization models, while models without constraints are called unconstrained models. It is worth noting that most real-world problems are constrained problems. Another classification is based on the type of variable. Variables can either be discrete or continuous. This therefore translates into discrete or continuous mathematical optimization problems. Yet another classification is based on the linearity of the objective function or constraints. We therefore have linear optimization models (commonly referred to as linear programming) or nonlinear optimization models (nonlinear programming). Other common classifications of mathematical models include static or dynamic models, which distinguish optimization models that include time variance. There are also stochastic or deterministic mathematic models, which distinguish models that contain elements of randomness and uncertainty. We also have single-objective or multi-objective optimization models, which distinguish models, based on the number of objectives that the modeler wants to maximize or minimize. There are many mathematical optimization solution techniques and this has become an oft-researched scientific area. Figure 3.1 shows a high-level overview of different mathematical optimization solution techniques. In this work, we categorize these mathematical optimization solution approaches into classical mathematical optimization solution approaches/techniques and meta-heuristic solution approaches/techniques.

3.2

Classical Mathematical Optimization Solution Approaches

Classical mathematical optimization solution approaches can be divided further into two classes, based on their solution methodology, namely, gradient descent methods and direct search methods [2]. Both approaches are explained further in the following subsections.

3.2 Classical Mathematical Optimization Solution Approaches

39

Fig. 3.1 An overview of mathematical optimization techniques [1]

3.2.1

Gradient Descent-Based Methods

Gradient descent-based methods are generally regarded as more proficient than other methods. This approach leverages on calculus and the determination of the derivatives of the objective function and constraints to obtain solutions [2]. An overview of some gradient descent solution methods is given below:

3.2.1.1

Steepest Descent Method

The steepest descent method starts with a guess for the minimum point and determines the direction of the steepest point from the initial minimum point [2]. If the direction obtained is not equal to zero (i.e. convergence has not yet been reached), then a line search for the minimum point is done along the negative to derivative direction [2]. The minimum point obtained then becomes the current point and the process is repeated till a zero value is obtained (i.e. convergence is obtained).

40

3.2.1.2

3 Mathematical Optimization Modeling and Solution Approaches

Conjugate Gradient Method

The conjugate gradient method is a variant of the steepest descent method and works by modifying the search direction [2]. In this method, recursive equations are used to determine the search direction. Because of this, after a few iterations, the search direction develops linear dependence. The degree of this is determined by calculating the included angle of two successive search directions. In the event that this value is approximately zero, the process is started all over again [2].

3.2.1.3

Newton’s Method

The Newton method uses second-order derivatives to determine search directions for the minimum point [2]. Second-order Taylor’s series expansion is used to determine the value of the function around the current point and the global minima are denoted, as the where the Hessian matrix is positively semi-definite [2].

3.2.1.4

Variable-Metric Method

The variable-metric method uses first-order derivatives to approximate the inverse of the Hessian matrix, thereby bypassing computation of the Hessian matrix and its inverse.

3.2.2

Direct Search Methods

Direct search methods do not use the function’s partial derivatives in determining the search direction. The search for the minimum is only conducted using the value of the function at different points, thus this method is usually suitable for less complicated problems with relatively few variables [2]. An overview of some gradient descent solution methods is given below:

3.2.2.1

Hooke-Jeeves Method

The Hooke-Jeeves method is made up of two moves: the exploratory and the pattern move [2]. The exploratory move is first done in the region around the current point to determine the best point, after which the pattern search is conducted using the two previously identified points.

3.2 Classical Mathematical Optimization Solution Approaches

3.2.2.2

41

Powell’s Conjugate Direction Method

Powell’s method uses results from prior solutions to develop new search directions. It uses the history of previous searches to develop a number of linearly independent directions and conducts searches in these directions [2]. It is an efficient search method.

3.3

Meta-Heuristic Methods

Meta-heuristic methods are indicated in Fig. 3.2 before being discussed in more detail.

Fig. 3.2 An overview of meta-heuristic optimization solution methods [3]

42

3.3.1

3 Mathematical Optimization Modeling and Solution Approaches

Genetic Algorithm

A genetic algorithm is a search algorithm that mimics the theory of genetics and natural selection from human nature as conceptualized by Darwin’s theory of evolution. The algorithm works by first randomly initializing an initial population of candidate solutions (with alterable chromosomes) [4]. The fitness level of this initial generation is evaluated. This population is thereafter refined first by the reproduction operator, which selects the best solutions from the initial random population and forms the mating pool. Once again, the fitness of this generation is evaluated. The cross-over operator is then used on solutions in the mating pool, after which the mutation operator is further used to obtain another generation whose fitness is yet again evaluated. The process continues until either the prerequisite fitness level is obtained or the maximum number of generations has been obtained.

3.3.2

Particle Swarm Optimization

Particle swarm optimization is a meta-heuristic optimization solution methodology that is regarded as a heuristic optimization procedure based on the movement of birds flocking or fish schooling [4]. Each individual (in either the bird flock or fish school) is referred to as a particle with its own position and velocity. These particles then move within a search space with guidance provided by the knowledge of each one’s own best position and the best position of the entire group. Knowledge of these two positions, known as local best and global best, respectively, enables the best solutions to be obtained.

3.3.3

Differential Evolution

Differential evolution is a meta-heuristic solution approach that begins with an initial population of potential solutions often referred to as agents [4]. The agents then obtain new positions in the search space by the weighted difference vector of existing agents in the population. This new position is evaluated and if its score (in relation to a fitness function) is an improvement on its previous position, it is included in the new population, otherwise the new agent is excluded from the new population. In this way, the position of the new population is an improvement on the former. This process is repeated till the maximum number of populations is obtained or a manageable fitness level is obtained.

3.3 Meta-Heuristic Methods

3.3.4

43

Group Search Optimization

Group search optimization is a meta-heuristic solution methodology that uses concepts related to the way animals search for food [4]. The initial population is selected and each member of this population can be a producer, scrounger or a ranger and mimics the way a herd of animals produces, scrounges or disperses (ranges) in a bid for sustenance. Each member of the population is thereafter evaluated based on its activities and its resultant position in relation to the fitness function. The process continues until a predefined termination criterion is obtained (Fig. 3.3).

Fig. 3.3 Flow chart for group search optimization [5]

44

3.3.5

3 Mathematical Optimization Modeling and Solution Approaches

Cuckoo Search

The cuckoo search method is a meta-heuristic solution methodology that is motivated by the way cuckoo birds brood their offspring [4]. Cuckoo birds lay one egg at a time and deposit the egg in a nest (the choice of this nest is random). An egg in a nest depicts a solution. After a while nests start filling up, with each new egg depicting a new solution. The nests with the largest collection of good quality eggs are carried over into the next generation and nests without good eggs are discarded. This process is continued until the predetermined termination criteria are reached.

3.3.6

Grey Wolf Optimization

This solution methodology is an algorithm that is based on the societal norms and hunting behavior of a pack of grey wolves [4]. In terms of societal hierarchy, grey wolves have alpha, beta, delta, and omega wolves in order of hierarchy. This societal hierarchy is adhered to when they hunt prey. The hunting activities of the pack consist of identifying a prey, encircling, and attacking the prey. The wolves in the pack are assumed to be potential solutions, with the hierarchy also determining the quality of solutions. Thus, the alpha wolves lead the pack to search for the prey, which, after it has been identified, is pursued and encircled. At that point the search process ends and the attack begins. The positions of all the solutions (wolves) are updated throughout the hunting procedure and their fitness is evaluated in relation to the fitness (location) of the prey. This process continues till the predetermined termination criteria have been reached.

3.3.7

Harmony Search Algorithm

The harmony search algorithm is a meta-heuristic solution algorithm based on the way musicians improve the harmony of their music [4]. A musician improves the harmony of his music by playing from memory (a harmony previously memorized), adjusting the pitch (modifying an existing memorized harmony) or making a randomized attempt. If the harmony is a candidate solution to a problem, then an optimal solution (perfect harmony) is obtained by initially obtaining previously memorized harmonies (solutions) and then modifying/adjusting them and finally comparing them with the randomized harmonics. Depending on which harmonics are better, the iteration continues. The aim of randomized harmonies is to force a search of space further afield. This process continues till a predetermined termination criterion is obtained.

3.3 Meta-Heuristic Methods

3.3.8

45

Bee Colony Optimization

Bee colony optimization is a meta-heuristic optimization methodology that mimics the way bees forage for food [4]. The bees are denoted as solutions; there are two types of bees: scouting and employed bees, with scouting bees searching for new food sources while employed bees obtain food from previously established food sources. Both classes of bees update themselves with information about their positions and abandon locations where food is not found. This process continues until either food is found or a predetermined termination criterion is achieved.

3.4

The Mathematical Modeling Process

The optimization modeling process is a cyclical process involving a number of steps. The primary and most critical step is understanding the nature of the real-world problem and the various factors (controllable or not) that influence the problem. What needs to be minimized/maximized and what are the factors that influence this quantity to be minimized/maximized? What factors can be ignored/ eliminated or approximated? The second step is the mathematical representation of the previously understood real-world problem. This involves confirming that the mathematical model obtained is a correct representation of the problem and any assumptions are valid/correct without distracting from the physical problem at hand [6, 7]. The third step involves obtaining data and processing the data that are required. This is followed by representing the mathematical model in syntax that the solver can understand, running the solver, and obtaining results. It is envisaged that the results should ideally be optimal. These results are subsequently analyzed in order to glean actionable intelligence from the results. The modeling process just described is an ideal process where everything goes according to plan, which hardly happens in modeling practice. In practice, a number of bottlenecks often occur, which will lead to a revision of some of the modeling assumptions, a review of data used, and/or refinement of algorithms or solvers deployed. This makes modeling and optimization of mathematical modeling a very cumbersome process. In practice, these are the steps to be used when building and solving a mathematical optimization model: i. Problem conceptualization The first step is problem conceptualization, which involves determining the problem, factors that influence the problem, and the mathematical abstraction of the problem.

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3 Mathematical Optimization Modeling and Solution Approaches

ii. Mathematical model design This step involves the conversion of the mathematical problem that has been determined into a form that can be solved by the computer. This is dependent on the platform used, solver, etc. iii. Results and analysis After running the software or solver and obtaining results, the results are analyzed to determine their accuracy and reasonableness. iv. Review After analyzing the results, especially in cases where shortfalls occur, the model is reviewed and appropriate corrective action is taken. If the results are accurate and reasonable, actionable intelligence is gleaned from the work.

3.5

Algebraic Modeling Languages

AMLs are advanced software platforms that enable the solution of large-scale complex optimization problems. AMLs are a bridge between the high-level algebraic mathematical representation of optimization problems and the low-level algorithms that enable their solution. AMLs use a syntax that is analogous to the mathematical symbolization of optimization problems, thereby substantially eliminating human effort and time in the optimization modeling process [8]. This introduces efficiency into the modeling process and more importantly, allows separation of the mathematical model, the data, and the solver. This separation allows for a number of different experiments and is one of the key concepts that have led to increased use of AMLs. Thus, the mathematical model and solver can be kept constant and the data can be scaled either up or down to represent larger or more realistic scenarios. In addition, the mathematical model and the date can be left constant and different solvers can experimented with in a bid to obtain the true optimum. The final choice of solvers used can be based on accuracy (true optimality), processing speed, and other relevant performance metrics.

3.5.1

Software

AMLs are high-level modeling platforms that can be used on computers. They contain integrated development environments and data exchange capabilities and are connected to a large number of algorithms that can be used to solve optimization

3.5 Algebraic Modeling Languages

47

problems. These algorithms are typically referred to as solvers and can either be open-source or commercial. AML software can be installed on most types of computers and operating systems and are deployed across a wide range of industries and businesses. There are many AMLs. However, five are adjudged to be the most important ones [9]: • Advanced Interactive Multidimensional Modeling System (AIMMS), founded by Johannes Bisschop • General Algebraic Modeling System, founded by Johannes Bisschop and Alexander Meeraus • Mathematical Programming Language, founded by Bjarni Kristjansson • A Mathematical Programming Language, founded by Robert Fourer, David Gay, and Brian Kernighan • Linear, Interactive, and Discrete Optimizer, founded by Kevin Cunningham and Linus Schrage. Most AMLs are quite similar in design philosophy; however, differences exist in their syntax, declarative and procedural elements, solver links, etc. AMLs can also be integrated with conventional mathematical and analytical platforms such as MATLAB and Microsoft Excel [8]. AMLs can also be deployed in software as a service framework. An example is the Network-enabled Optimization System server, which gives mathematical modelers access to a large number of solvers [8]. The server allows for a solution of 12 different kinds of optimization problems. This unique service allows for the sharing of optimization resources across a wide range of users. As stated before, algorithms that can be used to solve optimization problems are referred to as solvers. Solvers typically differ based on the type of optimization problems they can solve and whether they are free, commercial, or open-source. It is also possible to merge solvers, depending on the nature and complexity of the problem. Examples of prominent solvers include Sparse Nonlinear Optimizer, Computational Infrastructure for Operations Research, and IBM ILOG CPLEX Optimization Studio (CPLEX), among many others.

3.6

Advanced Interactive Multidimensional Modeling System

The AIMMS is an AML that has evolved into a prescriptive analytics platform that enables the creation of business-ready optimization applications. [10], [11]. It has also evolved to the AIMMS supply chain navigator, which provides applications for

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3 Mathematical Optimization Modeling and Solution Approaches

supply chain purposes. The software can solve a wide variety of optimization problems and includes an equally wide range of solvers [12]. A very attractive feature for academic institutions is the free academic license for academic institutions, which enables academics and students to access the software for research purposes.

3.6.1

Software Installation

AIMMS can be installed on various operating systems (Windows or Linux) and has many releases [10]. Except for academics and students, as stated, a license is required to use AIMMS. The process is intuitive and easy to follow. AIMMS has a number of helpful tutorials for varying levels of expertise.

3.6.2

AIMMS Tutorials

The AIMMS tutorial for beginners, as the name implies, is for beginners and introduces AIMMS [11]. It shows how to build relatively fast yet powerful models and includes useful data-handling functionality. After this tutorial, there are the AIMMS tutorials for professionals, which cover advanced topics for experienced modelers, including rolling horizons, etc. Practicalissuesto be addressed when using AIMMS: i. ii. iii. iv. v. vi. vii. viii.

Defining the sets and indices; Determining the parameters; Determining the decision variables; Defining the objective function Defining constraints; Building the model on the chosen AML; Loading the data; Choosing the solver to use (AIMMS does this automatically but the user can still choose solvers); ix. Obtaining results; x. Analyzing results; xi. Reviewing previously performed steps if results are not as expected.

In the next section, we briefly show how to use AIMMS for building retrofit energy efficiency programs, mentioned in 2.2.1, appliance scheduling using

3.6 Advanced Interactive Multidimensional Modeling System

49

TOU DR, mentioned in 2.3.1.5, and the game theory demand response programs, mentioned in 2.4.

3.7

Building Retrofit Energy Efficiency Programs Using AIMMS

The problem was introduced in Sect. 2.2.1. A complete description is provided in [13]. Set Definition: j represents customers and i represents technologies to be retrofitted. Variable Definition: nji is the number of units of technology i to be replaced by the customer j. Parameter Definition: Ri

is the rebate paid by the utility for each technology i in South African rands (ZAR). nji min and nji max are the minimum and maximum number of units of technology i to be replaced by the customer j. Ci is the average customer cost for replacing technology i (ZAR). DPi is the difference in power rating between the old technology and replacement technology (W). kj is the electricity tariff for the customer j in ZAR/kWh. hji is the average number of hours that a customer j uses technology i in a day. MR is the maximum rebate Eskom is paying for any project capped at ZAR 750,000, i.e., MR = 750,000. CBj is the maximum amount of money customer j is willing to spend on replacing inefficient technologies. J is the total number of customers. I is the maximum number of technologies the customer selects from the standard product program and is willing to change.

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3 Mathematical Optimization Modeling and Solution Approaches

Objective Function and Constraints maxf2 ¼

J X I X

½nji ðRi  Ci Þ þ 365nji hji DPi kj 

ð3:4Þ

j¼1 i¼1

subject to: I X

nji Ri  MR;

ð3:5Þ

nji Ci  CBj ;

ð3:6Þ

i¼1 I X i¼1

1kW 

I X

nji DPi  250kW;

ð3:7Þ

i¼1 I X

365nji hji DPi  2MWh:

ð3:8Þ

i¼1

for all j = 1,2,. .. .. ., J. The following is a brief explanation of the constraints [9]: • The first constraint (3.5) ensures that for each customer, the rebate paid is limited to ZAR750 000. • The second constraint (3.6) limits the total customer investment to the individual customer’s budgeted amount. • The third constraint (3.7) ensures that the demand reduction for each customer due to the technology replacement is between 1 kW and 250 kW. • The fourth constraint (3.8) compels the energy saved per annum to be greater than 2 MWh.

3.7 Building Retrofit Energy Efficiency Programs Using AIMMS Model Main_Example3.7 { Set customers { Index: j; } Set technologies { Index: i; } Parameter Rebate { IndexDomain: (i); } Parameter customercost { IndexDomain: (i); } Parameter Poweratingdiff { IndexDomain: (i); } Parameter Tarriff { IndexDomain: (j); } Parameter hourofuse { IndexDomain: (j,i); } Parameter MaxRebate; Parameter Max_Replacementcost { IndexDomain: (j); } Parameter Min_technologyunit { IndexDomain: (j,i); } Parameter Max_technologyunit { IndexDomain: (j,i); } Variable no_technologyunit { IndexDomain: (j,i); Range: nonnegative; } Variable Obj_Function { Range: free;

51

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3 Mathematical Optimization Modeling and Solution Approaches

Definition: sum((j,i),(no_technologyunit(j,i)*(Rebate(i)customercost(i))))+sum((j,i),(365*(no_technologyunit(j,i)*(hourofuse(j,i)*(Poweratingd iff(i))*(Tarriff(j)))))); Comment: "sum((j,i),((no_technologyunit(j,i)*(Rebate(i)customercost(i)))+(365(no_technologyunit(j,i)*(hourofuse(j,i)*(Poweratingdiff(i))*(Tar riff(j)))))))"; } Constraint constr1 { IndexDomain: (j); Definition: sum(i,(no_technologyunit(j,i))*(Rebate(i)))